Revision NotesExam QuestionsFlashcardsPast PapersMock Exams
IBMathsDPAnalysis & Approaches (AA)SLRevision Notes5. CalculusFurther IntegrationArea Between 2 Curves

Area Between 2 Curves (DP IB Analysis & Approaches (AA)): Revision Note

Paul

Written by: Paul

Reviewed by: Dan Finlay

Updated on

DP Analysis & Approaches (AA): SLDP Analysis & Approaches (AA): HL

Did this video help you?

Area Between 2 Curves

What do we mean by 'area between two curves'?

  • Areas whose boundaries include two curves can be found by integration

    • The area between two curves will be the difference of the areas under the two curves

      • both areas will require a definite integral

    • Finding points of intersection may involve a more awkward equation than solving for a curve and a line

Graph showing curves y = f(x) and y = g(x) intersecting at 3 points with x-coordinates a, b and c. Shaded areas R_1 and R_2 are enclosed by the two curves. The area of R_1, where f(x) is above g(x), is given by the integral of (f(x)-g(x)) between a and b. The area of R_2, where g(x) is above f(x), is given by the integral of (g(x)-f(x)) between b and c.

How do I find the area between two curves?

  • STEP 1

    If not given, sketch the graphs of both curves on the same diagram

    • You can use a GDC to help with this step
       

  •  STEP 2

    Find the intersections of the two curves

    • If no diagram is given this will help identify the area(s) to be found

  • STEP 3

    For each area (there may only be one) determine which curve is the ‘upper’ boundary

    For each area, write a definite integral of the formspace integral subscript a superscript b open parentheses y subscript 1 minus y subscript 2 close parentheses space straight d x

    • wherespace y subscript 1 is the function for the ‘upper’ boundary andspace y subscript 2 is the function for the ‘lower’ boundary

Examiner Tips and Tricks

Be careful when there is more than one region – the ‘upper’ and ‘lower’ boundaries will often switch between regions!

  •  STEP 4

    Evaluate the definite integrals and sum them up to find the total area

Examiner Tips and Tricks

As always, sketching a diagram, or adding info to a diagram that is given, is very helpful in questions like this. On a calculator paper you can use your GDC to help.

Also note that you don't have to worry about areas being below the x-axis with area between two curves. As long as you have the 'upper' and 'lower' curves the right way round, open parentheses y subscript 1 minus y subscript 2 close parentheses inside the integrals will always be positive. This means you can never have a 'negative integral'.

Worked Example

The diagram below shows the curves with equationsspace y equals f left parenthesis x right parenthesis and space y equals g left parenthesis x right parenthesis where

space f left parenthesis x right parenthesis equals left parenthesis x minus 2 right parenthesis left parenthesis x minus 3 right parenthesis squared space and space g left parenthesis x right parenthesis equals x squared minus 5 x plus 6.

Find the area of the shaded region.

5-4-4-ib-sl-aa-only-we3-qu-img
5-4-4-ib-sl-aa-only-we3-soltn

You've read 0 of your 5 free revision notes this week

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourselfFlashcards

    Did this page help you?

    Previous:Area Between a Curve and a LineNext:Modelling with Differentiation
    Author
    Reviewer
    Paul

    Author: Paul

    Expertise: Maths Content Creator

    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.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.