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Areas Between Curves (Edexcel IGCSE Further Pure Maths)

Revision Note

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Area Between a Curve and a Line

What do we mean by 'area between a curve and a line'?

  • Areas whose boundaries include a curve and a (non-vertical) straight line can be found using integration

    • For an area under a curve a definite integral will be needed

    • For an area under a line the shape formed will be a trapezium or triangle

      • basic area formulae can be used rather than a definite integral

      • (although a definite integral would still work)

  • The area required could be the sum or difference of areas under the curve and line

Sum of two areas under a curve and a line
Difference between two areas below a curve and a line

How do I find the area between a curve and a line?

  • STEP 1
    If not given, sketch the graphs of the curve and line on the same diagram

  • STEP 2
    Find the intersections of the curve and the line

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

  • STEP 3
    Determine whether the area required is a sum or difference

    • Calculate the area under a curve using a integral of the form integral subscript a superscript b y space straight d x

    • Calculate the area under a line using

      • space A equals 1 half b h for a triangle

      • space A equals 1 half left parenthesis a plus b right parenthesis h for a trapezium

        • For a trapezium, y-coordinates will be needed for a  and  b

        • and the height  h will lie parallel to the x-axis

    • Those areas will need to be added or subtracted, depending on the question

  • STEP 4
    Evaluate the definite integral(s)

    • Then find the sum or difference of areas as required

Examiner Tips and Tricks

  • Add information to any diagram provided

    • intersections between lines and curves

    • mark and shade the area you’re trying to find

  • If no diagram is provided, sketch one!

Worked Example

The regionspace R is bounded by the curve with equation y equals 10 x minus x squared minus 16 and the line with equationspace y equals 8 minus x.

(a) Sketch the graphs of the curve and the line on the same set of axes.

Be sure to Identify and label the regionspace R on your sketch, and indicate the points of intersection between the curve and the line.  You may assume without proof that the curve's x-axis intercepts all lie on the positive x-axis.

Because of the minus sign in front of x squared, the curve will be an 'upside down u-shaped' parabola

We need to find the points of intersection of the curve and line
Set their equations equal and solve to find the x-coordinates

table attributes columnalign right center left columnspacing 0px end attributes row cell 8 minus x end cell equals cell 10 x minus x squared minus 16 end cell row cell x squared minus 11 x plus 24 end cell equals 0 row cell open parentheses x minus 3 close parentheses open parentheses x minus 8 close parentheses end cell equals 0 end table


x equals 3 space space or space space x equals 8


So the curve and line intersect when x equals 3 and when x equals 8

Substitute into the equation of the line to find the corresponding y-coordinates

x equals 3 colon space space space y equals 8 minus 3 equals 5 x equals 8 colon space space space y equals 8 minus 8 equals 0


So the points of intersection are open parentheses 3 comma space 5 close parentheses and open parentheses 8 comma space 0 close parentheses

That gives us enough information to sketch the graphs

Area formed between a curve and line

(b) Find the area of region R

Here we're going to need a difference of areas:  (area under curve)minus(area under line)


The area under the line is a right-angled triangle with vertices at open parentheses 3 comma space 0 close parentheses, open parentheses 3 comma space 5 close parentheses and open parentheses 8 comma space 0 close parentheses

So the height is 5 minus 0 equals 5 and the base is 8 minus 3 equals 5

Area space of space triangle equals 1 half cross times 5 cross times 5 equals 25 over 2


For the area under a curve we need to use a definite integral between x equals 3 and x equals 8

table row cell Area space under space the space curve end cell equals cell integral subscript 3 superscript 8 open parentheses 10 x minus x squared minus 16 close parentheses space straight d x end cell row blank equals cell open square brackets 5 x squared minus 1 third x cubed minus 16 x close square brackets subscript 3 superscript 8 end cell row blank equals cell open parentheses 5 open parentheses 8 close parentheses squared minus 1 third open parentheses 8 close parentheses cubed minus 16 open parentheses 8 close parentheses close parentheses minus open parentheses 5 open parentheses 3 close parentheses squared minus 1 third open parentheses 3 close parentheses cubed minus 16 open parentheses 3 close parentheses close parentheses end cell row blank equals cell open parentheses 320 minus 512 over 3 minus 128 close parentheses minus open parentheses 45 minus 9 minus 48 close parentheses end cell row blank equals cell 64 over 3 minus open parentheses negative 12 close parentheses end cell row blank equals cell 100 over 3 end cell end table


For the area of R, subtract the area of the triangle from the area under the curve

Area space of space region space R equals 100 over 3 minus 25 over 2 equals 125 over 6


bold 125 over bold 6 bold space bold units to the power of bold 2

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

Areas formed between two curves

How do I find the area between two curves?

  • STEP 1
    If not given, sketch the graphs of both curves on the same diagram 

  • STEP 2
    If not given, find the intersections of the two curves

    • These are needed to identify the area(s) to be calculated

    • and also to set up the correct integrals

  • STEP 3
    Determine which curve is the ‘upper’ boundary for each region

    • For each region, the area is given by definite integral of the formspace integral subscript a superscript b left parenthesis y subscript 1 minus y subscript 2 right parenthesis space straight d x

      • y subscript 1 is the function forming the ‘upper’ boundary

      • y subscript 2 is the function forming the ‘lower’ boundary

    • Be careful when there is more than one region

      • Which functions form the ‘upper’ and ‘lower’ boundaries can change

  • STEP 4
    Evaluate the definite integral(s)

    • If there is more than one region, add their areas together to find the total area

    • As long as 'y subscript 1' in the integrals is always the upper function

      • then  y subscript 1 minus y subscript 2 greater or equal than 0

      • This means you don't have to worry about negative integrals

      • even if part or all of the area between the curves is below the x-axis

Examiner Tips and Tricks

  • Add information to any given diagram as you work through a question

    • intersections between curves

    • mark and shade the area you're trying to find

  • If no diagram is provided sketch one

Worked Example

The diagram below shows the curves with equations y equals straight f left parenthesis x right parenthesis and y equals straight g left parenthesis x right parenthesis, where  straight f left parenthesis x right parenthesis equals left parenthesis x minus 2 right parenthesis left parenthesis x minus 3 right parenthesis squared  and  straight g left parenthesis x right parenthesis equals x squared minus 5 x plus 6.

Find the area of the shaded region.

Areas formed between a cubic and a quadratic


Start by finding the points of intersection

Set the equations of the curves equal to each other, and solve to find the x-coordinates

open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses squared equals x squared minus 5 x plus 6


It's tempting to expand the brackets on the left-hand side of the equation
Actually. here it will be more useful to factorise the right-hand side

table row cell open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses squared end cell equals cell open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses end cell row cell open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses squared minus open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses end cell equals 0 row cell open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses open parentheses open parentheses x minus 3 close parentheses minus 1 close parentheses end cell equals 0 row cell open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses open parentheses x minus 4 close parentheses end cell equals 0 end table

x equals 2 comma space 3 space or space 4


Note that we don't need to know the corresponding y-coordinates here!

We have two regions here

The first region is from x equals 2 to x equals 3, and straight f open parentheses x close parentheses is the 'upper curve'

table row cell Region space 1 end cell equals cell integral subscript 2 superscript 3 open parentheses open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses squared minus open parentheses x squared minus 5 x plus 6 close parentheses close parentheses space straight d x end cell row blank equals cell integral subscript 2 superscript 3 open parentheses x cubed minus 9 x squared plus 26 x minus 24 close parentheses space straight d x end cell row blank equals cell open square brackets 1 fourth x to the power of 4 minus 3 x cubed plus 13 x squared minus 24 x close square brackets subscript 2 superscript 3 end cell row blank equals cell open parentheses 1 fourth open parentheses 3 close parentheses to the power of 4 minus 3 open parentheses 3 close parentheses cubed plus 13 open parentheses 3 close parentheses squared minus 24 open parentheses 3 close parentheses close parentheses minus open parentheses 1 fourth open parentheses 2 close parentheses to the power of 4 minus 3 open parentheses 2 close parentheses cubed plus 13 open parentheses 2 close parentheses squared minus 24 open parentheses 2 close parentheses close parentheses end cell row blank equals cell open parentheses 81 over 4 minus 81 plus 117 minus 72 close parentheses minus open parentheses 4 minus 24 plus 52 minus 48 close parentheses end cell row blank equals cell negative 63 over 4 minus open parentheses negative 16 close parentheses end cell row blank equals cell 1 fourth end cell end table


The second region is from x equals 3 to x equals 4 and straight g open parentheses x close parentheses is the 'upper curve'

table row cell Region space 2 end cell equals cell integral subscript 3 superscript 4 open parentheses open parentheses x squared minus 5 x plus 6 close parentheses minus open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses squared close parentheses space straight d x end cell row blank equals cell integral subscript 3 superscript 4 open parentheses negative x cubed plus 9 x squared minus 26 x plus 24 close parentheses space straight d x end cell row blank equals cell open square brackets negative 1 fourth x to the power of 4 plus 3 x cubed minus 13 x squared plus 24 x close square brackets subscript 3 superscript 4 end cell row blank equals cell open parentheses negative 1 fourth open parentheses 4 close parentheses to the power of 4 plus 3 open parentheses 4 close parentheses cubed minus 13 open parentheses 4 close parentheses squared plus 24 open parentheses 4 close parentheses close parentheses minus open parentheses negative 1 fourth open parentheses 3 close parentheses to the power of 4 plus 3 open parentheses 3 close parentheses cubed minus 13 open parentheses 3 close parentheses squared plus 24 open parentheses 3 close parentheses close parentheses end cell row blank equals cell open parentheses negative 64 plus 192 minus 208 plus 96 close parentheses minus open parentheses negative 81 over 4 plus 81 minus 117 plus 72 close parentheses end cell row blank equals cell 16 minus 63 over 4 end cell row blank equals cell 1 fourth end cell end table


Now just add the two areas together to get the total area

Total space area equals 1 fourth plus 1 fourth equals 1 half

Area of the shaded region bold equals bold 1 over bold 2 bold space bold units to the power of bold 2

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths 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.