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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 5. Calculus / 5.9 Advanced Integration / 5.9.4 Advanced Applications of Integration


5.9.4 Advanced Applications of Integration


Area Between Curve & y-axis

What is meant by the area between a curve and the y-axis?

5-9-4-ib-hl-ai-aa-extraaa-fig1-area-yaxis

 

  • The area referred to is the region bounded by
    • the graph of y equals f left parenthesis x right parenthesis
    • the y-axis
    • the horizontal line y equals a
    • the horizontal line y equals b
  • The exact area can be found by evaluating a definite integral
  • The graph of y equals f left parenthesis x right parenthesis could be a straight line
    • using basic shape area formulae may be easier than integration
      • e.g. area of a trapezium: A equals 1 half h open parentheses a plus b close parentheses

How do I find the area between a curve and the y-axis?

  • Use the formula

A equals integral subscript a superscript b open vertical bar x close vertical bar space d x

    • This is given in the formula booklet
    • The function is normally given in the form y equals f left parenthesis x right parenthesis
      • so will need rearranging into the form x equals g left parenthesis y right parenthesis
    • a and b may not be given directly as could involve the x-axis (y equals 0) and/or a root of x equals g left parenthesis y right parenthesis
      • use a GDC to plot the curve, sketch it and highlight the area to help

STEP 1
Identify the limits a and b
Sketch the graph of y equals f left parenthesis x right parenthesis or use a GDC to do so, especially if a and b are not given directly in the question

STEP 2
Rearrange y equals f left parenthesis x right parenthesis into the form x equals g left parenthesis y right parenthesis
This is similar to finding the inverse function f to the power of negative 1 end exponent left parenthesis x right parenthesis

STEP 3
Evaluate the formula to evaluate the integral and find the area required
If using a GDC remember to include the modulus ( | … | ) symbols around x 

  • In trickier problems some (or all) of the area may be ‘negative’
    • this will be any area that is left of the y-axis (negative x-values)
    • |x| makes such areas ‘positive’
      • a GDC will apply ‘|x|’ automatically as long as the | … | are included
      • otherwise, to apply ‘|x|’, split the integral into positive and negative parts; write an integral and evaluate each part separately and add the modulus of each part together to give the total area

Exam Tip

  • Longer problems may require you to rotate an area around both the x-axis and the y-aixs
  • Sketch and/or use your GDC to help visualise what the problem looks like

Worked Example

Find the area enclosed by the curve with equation y equals 2 plus square root of x plus 4 end root, the y-axis and the horizontal lines with equations y equals 3 and y equals 6.

5-9-4-ib-hl-ai-aa-extraaa-we1-soltn

Volumes of Revolution Around x-axis

What is a volume of revolution around the x-axis?

  • A solid of revolution is formed when an area bounded by a function y equals f left parenthesis x right parenthesis
    (and other boundary equations) is rotated 2 pi radians left parenthesis 360 degree right parenthesis around the x-axis
  • The volume of revolution is the volume of this solid

  • Be careful – the ’front’ and ‘back’ of this solid are flat
    • they were created from straight (vertical) lines
    • 3D sketches can be misleading

How do I solve problems involving the volume of revolution around x-axis?

  • Use the formula

 V equals pi integral subscript a superscript b y squared space d x

  • This is given in the formula booklet
  • y is a function of x
  • x equals a and x equals b are the equations of the (vertical) lines bounding the area
    • If x equals a and x equals b are not stated in a question, the boundaries could involve the y-axis (x equals 0) and/or a root of y equals f left parenthesis x right parenthesis
    • Use a GDC to plot the curve, sketch it and highlight the area to help
  • Visualising the solid created is helpful
    • Try sketching some functions and their solids of revolution to help

STEP 1
Identify the limits a and b
Sketching the graph of y equals f left parenthesis x right parenthesis or using a GDC to do so is helpful, especially when a and b are not given directly in the question

STEP 2
Square y

STEP 3
Use the formula to evaluate the integral and find the volume of revolution
An answer may be required in exact form

Exam Tip

  • If the given function involves a square root(s), problems can seem quite daunting
    • However, this is often deliberate, as the square root will be squared when applying the Volume of Revolution formula, and should leave the integrand as something more manageable
  • Whether a diagram is given or not, using your GDC to plot the curve, limits, etc (where possible) can help you to visualise and make progress with problems

Worked Example

Find the volume of the solid of revolution formed by rotating the region bounded by the graph of y equals square root of 3 x squared plus 2 end root, the coordinate axes and the line x equals 3 by 2 pi radians around the x-axis.  Give your answer as an exact multiple of pi.

5-9-4-ib-hl-ai-aa-extraaa-we2-soltn

Volumes of Revolution Around y-axis

What is a volume of revolution around the y-axis?

  • Very similar to above, this is a solid of revolution which is formed when an area bounded by a function y equals f left parenthesis x right parenthesis (and other boundary equations) is rotated 2 pi radians left parenthesis 360 degree right parenthesis around the y-axis
  • The volume of revolution is the volume of this solid

How do I solve problems involving the volume of revolution around y-axis?

  • Use the formula

V equals straight pi integral subscript a superscript b x squared space d y 

  • This is given in the formula booklet
  • The function is usually given in the form y equals f left parenthesis x right parenthesis
    • so will need rearranging into the form x equals g left parenthesis y right parenthesis
  • a and b may not be given directly as could involve the x-axis (y equals 0) and/or a root of x equals g left parenthesis y right parenthesis
    • Use a GDC to plot the curve, sketch it and highlight the area to help
  • Visualising the solid created is helpful

STEP 1
Identify the limits a and b
Sketching the graph of y equals f left parenthesis x right parenthesis or using a GDC to do so is helpful, especially if a and b are not given directly in the question

STEP 2
Rearrange y equals f left parenthesis x right parenthesis into the form x equals g left parenthesis y right parenthesis
This is similar to finding the inverse function f to the power of negative 1 end exponent left parenthesis x right parenthesis

STEP 3
Square x

STEP 4
Use the formula to evaluate the integral and find the volume of revolution
An answer may be required in exact form

Exam Tip

  • If the given function involves a square root, problems can seem quite daunting
    • This is often deliberate, as the square root will be squared when applying the Volume of Revolution formula and the integrand will then become more manageable
  • Whether a diagram is given or not, using your GDC to plot the curve, limits, etc (where possible) can help you to visualise the problem and make progress

Worked Example

Find the volume of the solid of revolution formed by rotating the region bounded by the graph of y equals arcsin space open parentheses 2 x plus 1 close parentheses and the coordinate axes by 2 straight pi radians around the y-axis.  Give your answer to three significant figures.

yzeHWPTm_5-9-4-ib-hl-ai-aa-extraaa-we3-soltn



  • 1. Number & Algebra
    • 1.1 Number & Algebra Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
          • 1.1.3 Partial Fractions
          • 1.2 Exponentials & Logs
            • 1.2.1 Introduction to Logarithms
              • 1.2.2 Laws of Logarithms
                • 1.2.3 Solving Exponential Equations
                • 1.3 Sequences & Series
                  • 1.3.1 Language of Sequences & Series
                    • 1.3.2 Arithmetic Sequences & Series
                      • 1.3.3 Geometric Sequences & Series
                        • 1.3.4 Applications of Sequences & Series
                          • 1.3.5 Compound Interest & Depreciation
                          • 1.4 Simple Proof & Reasoning
                            • 1.4.1 Proof
                            • 1.5 Further Proof & Reasoning
                              • 1.5.1 Proof by Induction
                                • 1.5.2 Proof by Contradiction
                                • 1.6 Binomial Theorem
                                  • 1.6.1 Binomial Theorem
                                    • 1.6.2 Extension of The Binomial Theorem
                                    • 1.7 Permutations & Combinations
                                      • 1.7.1 Counting Principles
                                        • 1.7.2 Permutations & Combinations
                                        • 1.8 Complex Numbers
                                          • 1.8.1 Intro to Complex Numbers
                                            • 1.8.2 Modulus & Argument
                                              • 1.8.3 Introduction to Argand Diagrams
                                              • 1.9 Further Complex Numbers
                                                • 1.9.1 Geometry of Complex Numbers
                                                  • 1.9.2 Forms of Complex Numbers
                                                    • 1.9.3 Complex Roots of Polynomials
                                                      • 1.9.4 De Moivre's Theorem
                                                        • 1.9.5 Roots of Complex Numbers
                                                        • 1.10 Systems of Linear Equations
                                                          • 1.10.1 Systems of Linear Equations
                                                            • 1.10.2 Algebraic Solutions
                                                          • 2. Functions
                                                            • 2.1 Linear Functions & Graphs
                                                              • 2.1.1 Equations of a Straight Line
                                                              • 2.2 Quadratic Functions & Graphs
                                                                • 2.2.1 Quadratic Functions
                                                                  • 2.2.2 Factorising & Completing the Square
                                                                    • 2.2.3 Solving Quadratics
                                                                      • 2.2.4 Quadratic Inequalities
                                                                        • 2.2.5 Discriminants
                                                                        • 2.3 Functions Toolkit
                                                                          • 2.3.1 Language of Functions
                                                                            • 2.3.2 Composite & Inverse Functions
                                                                              • 2.3.3 Symmetry of Functions
                                                                                • 2.3.4 Graphing Functions
                                                                                • 2.4 Other Functions & Graphs
                                                                                  • 2.4.1 Exponential & Logarithmic Functions
                                                                                    • 2.4.2 Solving Equations
                                                                                      • 2.4.3 Modelling with Functions
                                                                                      • 2.5 Reciprocal & Rational Functions
                                                                                        • 2.5.1 Reciprocal & Rational Functions
                                                                                        • 2.6 Transformations of Graphs
                                                                                          • 2.6.1 Translations of Graphs
                                                                                            • 2.6.2 Reflections of Graphs
                                                                                              • 2.6.3 Stretches Graphs
                                                                                                • 2.6.4 Composite Transformations of Graphs
                                                                                                • 2.7 Polynomial Functions
                                                                                                  • 2.7.1 Factor & Remainder Theorem
                                                                                                    • 2.7.2 Polynomial Division
                                                                                                      • 2.7.3 Polynomial Functions
                                                                                                        • 2.7.4 Roots of Polynomials
                                                                                                        • 2.8 Inequalities
                                                                                                          • 2.8.1 Solving Inequalities Graphically
                                                                                                            • 2.8.2 Polynomial Inequalities
                                                                                                            • 2.9 Further Functions & Graphs
                                                                                                              • 2.9.1 Modulus Functions
                                                                                                                • 2.9.2 Modulus Transformations
                                                                                                                  • 2.9.3 Modulus Equations & Inequalities
                                                                                                                    • 2.9.4 Reciprocal & Square Transformations
                                                                                                                  • 3. Geometry & Trigonometry
                                                                                                                    • 3.1 Geometry Toolkit
                                                                                                                      • 3.1.1 Coordinate Geometry
                                                                                                                        • 3.1.2 Radian Measure
                                                                                                                          • 3.1.3 Arcs & Sectors
                                                                                                                          • 3.2 Geometry of 3D Shapes
                                                                                                                            • 3.2.1 3D Coordinate Geometry
                                                                                                                              • 3.2.2 Volume & Surface Area
                                                                                                                              • 3.3 Trigonometry Toolkit
                                                                                                                                • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                                                  • 3.3.2 Non Right-Angled Trigonometry
                                                                                                                                    • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                                                    • 3.4 Trigonometry
                                                                                                                                      • 3.4.1 The Unit Circle
                                                                                                                                        • 3.4.2 Exact Values
                                                                                                                                        • 3.5 Trigonometric Functions & Graphs
                                                                                                                                          • 3.5.1 Graphs of Trigonometric Functions
                                                                                                                                            • 3.5.2 Transformations of Trigonometric Functions
                                                                                                                                              • 3.5.3 Modelling with Trigonometric Functions
                                                                                                                                              • 3.6 Trigonometric Equations & Identities
                                                                                                                                                • 3.6.1 Simple Identities
                                                                                                                                                  • 3.6.2 Compound Angle Formulae
                                                                                                                                                    • 3.6.3 Double Angle Formulae
                                                                                                                                                      • 3.6.4 Relationship Between Trigonometric Ratios
                                                                                                                                                        • 3.6.5 Linear Trigonometric Equations
                                                                                                                                                          • 3.6.6 Quadratic Trigonometric Equations
                                                                                                                                                          • 3.7 Inverse & Reciprocal Trig Functions
                                                                                                                                                            • 3.7.1 Reciprocal Trig Functions
                                                                                                                                                              • 3.7.2 Inverse Trig Functions
                                                                                                                                                              • 3.8 Further Trigonometry
                                                                                                                                                                • 3.8.1 Trigonometric Proof
                                                                                                                                                                  • 3.8.2 Strategy for Trigonometric Equations
                                                                                                                                                                  • 3.9 Vector Properties
                                                                                                                                                                    • 3.9.1 Introduction to Vectors
                                                                                                                                                                      • 3.9.2 Position & Displacement Vectors
                                                                                                                                                                        • 3.9.3 Magnitude of a Vector
                                                                                                                                                                          • 3.9.4 The Scalar Product
                                                                                                                                                                            • 3.9.5 Geometric Proof with Vectors
                                                                                                                                                                            • 3.10 Vector Equations of Lines
                                                                                                                                                                              • 3.10.1 Vector Equations of Lines
                                                                                                                                                                                • 3.10.2 Applications to Kinematics
                                                                                                                                                                                  • 3.10.3 Pairs of Lines in 3D
                                                                                                                                                                                    • 3.10.4 The Vector Product
                                                                                                                                                                                      • 3.10.5 Shortest Distances with Lines
                                                                                                                                                                                      • 3.11 Vector Planes
                                                                                                                                                                                        • 3.11.1 Vector Equations of Planes
                                                                                                                                                                                          • 3.11.2 Intersections of Lines & Planes
                                                                                                                                                                                            • 3.11.3 Angles Between Lines & Planes
                                                                                                                                                                                              • 3.11.4 Shortest Distances with Planes
                                                                                                                                                                                            • 4. Statistics & Probability
                                                                                                                                                                                              • 4.1 Statistics Toolkit
                                                                                                                                                                                                • 4.1.1 Sampling & Data Collection
                                                                                                                                                                                                  • 4.1.2 Statistical Measures
                                                                                                                                                                                                    • 4.1.3 Frequency Tables
                                                                                                                                                                                                      • 4.1.4 Linear Transformations of Data
                                                                                                                                                                                                        • 4.1.5 Outliers
                                                                                                                                                                                                          • 4.1.6 Univariate Data
                                                                                                                                                                                                            • 4.1.7 Interpreting Data
                                                                                                                                                                                                            • 4.2 Correlation & Regression
                                                                                                                                                                                                              • 4.2.1 Bivariate Data
                                                                                                                                                                                                                • 4.2.2 Correlation & Regression
                                                                                                                                                                                                                • 4.3 Probability
                                                                                                                                                                                                                  • 4.3.1 Probability & Types of Events
                                                                                                                                                                                                                    • 4.3.2 Conditional Probability
                                                                                                                                                                                                                      • 4.3.3 Bayes' Theorem
                                                                                                                                                                                                                        • 4.3.4 Sample Space Diagrams
                                                                                                                                                                                                                        • 4.4 Probability Distributions
                                                                                                                                                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                                                                                                                                                            • 4.4.2 Mean & Variance
                                                                                                                                                                                                                            • 4.5 Binomial Distribution
                                                                                                                                                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                                                                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                                                                                                                • 4.6 Normal Distribution
                                                                                                                                                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                                                                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                                                                                                                                                      • 4.6.3 Standardisation of Normal Variables
                                                                                                                                                                                                                                      • 4.7 Further Probability Distributions
                                                                                                                                                                                                                                        • 4.7.1 Probability Density Function
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                  • 5.2.3 Higher Order Derivatives
                                                                                                                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                                                                                                                        • 5.3 Integration
                                                                                                                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                          • 5.7 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.7.1 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.8 Advanced Differentiation
                                                                                                                                                                                                                                                                              • 5.8.1 First Principles Differentiation
                                                                                                                                                                                                                                                                                • 5.8.2 Applications of Chain Rule
                                                                                                                                                                                                                                                                                  • 5.8.3 Implicit Differentiation
                                                                                                                                                                                                                                                                                    • 5.8.4 Differentiating Further Functions
                                                                                                                                                                                                                                                                                    • 5.9 Advanced Integration
                                                                                                                                                                                                                                                                                      • 5.9.1 Integrating Further Functions
                                                                                                                                                                                                                                                                                        • 5.9.2 Further Techniques of Integration
                                                                                                                                                                                                                                                                                          • 5.9.3 Integrating with Partial Fractions
                                                                                                                                                                                                                                                                                            • 5.9.4 Advanced Applications of Integration
                                                                                                                                                                                                                                                                                              • 5.9.5 Modelling with Volumes of Revolution
                                                                                                                                                                                                                                                                                              • 5.10 Differential Equations
                                                                                                                                                                                                                                                                                                • 5.10.1 Numerical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                  • 5.10.2 Analytical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.10.3 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.11 MacLaurin Series
                                                                                                                                                                                                                                                                                                      • 5.11.1 Maclaurin Series
                                                                                                                                                                                                                                                                                                        • 5.11.2 Maclaurin Series from Differential Equations
                                                                                                                                                                                                                                                                                                        • 5.12 Further Limits (inc l'Hôpital's Rule)
                                                                                                                                                                                                                                                                                                          • 5.12.1 Further Limits
                                                                                                                                                                                                                                                                                                        Paul Freeman

                                                                                                                                                                                                                                                                                                        Author: Paul

                                                                                                                                                                                                                                                                                                        Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.


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