GCSEMathsWJECMaths & Numeracy (Double Award)HigherRevision NotesAlgebraEstimating Gradients & Areas under GraphsFinding Areas under Graphs

Finding Areas under Graphs (WJEC GCSE Maths & Numeracy (Double Award)): Revision Note

Exam code: 3320

Jamie Wood

Written by: Jamie Wood

Reviewed by: Mark Curtis

Updated on

WJEC Maths & Numeracy (Double Award): HigherWJEC Eduqas Maths: HigherOCR Maths: HigherAQA Maths: HigherEdexcel Maths: Higher

Finding areas under graphs

How do I estimate the area under a graph?

  • To find an estimate for the area:

    • Split area into vertical strips

    • Draw straight lines between the tops of the strips

      • This turns them into trapeziums

    • Find the area of each strip (trapezium) using Area equals 1 half open parentheses a plus b close parentheses h

    • Add all the areas together

What is the trapezium rule?

  • The trapezium rule can speed up your working for finding the area under a graph

    • It works in the same way as finding the area of several trapeziums and adding them up

  • The formula for the trapezium rule is

    • Area equals 1 half open parentheses y subscript 0 plus 2 open parentheses y subscript 1 plus y subscript 2 plus... plus y subscript n minus 1 end subscript close parentheses plus y subscript n close parentheses cross times h

      • h is the width of each trapezium (along the x axis)

      • y subscript 0 to y subscript n are the y-values (the lengths of the sides of the trapeziums)

    • Substitute the values into the formula and you can find the area in one step

    • It may be easier to think of the formula as Area equals 1 half open parentheses first plus 2 open parentheses sum space of space the space middles close parentheses plus last close parentheses cross times h

How do I know if my answer is an underestimate or an overestimate?

  • A common exam question is to ask if your estimate of the area is an underestimate or an overestimate

  • To answer this, simply look at the straight lines joining the tops of your strips

    • If the straight lines are below the curve, it is an underestimate

    • If the straight lines are above the curve, it is an overestimate

  • In your exam, the lines will all be below or all be above the curve- though it may be difficult to tell which for some strips

Examiner Tips and Tricks

This is particularly useful when working with speed-time graphs that are curved. The area underneath a speed-time graph is equal to the distance travelled.

Worked Example

The graph below shows y equals cube root of x for 0 less or equal than x less or equal than 1

Graph showing a concave curve from the origin, increasing from y=0 at x=0 to y=1 at x=1, on a grid with labelled axes. This is the graph of the cube root of x.

Find an estimate for the area between the curve, the x axis and the line x equals 1. Use four strips of equal width.

Answer:

Split the area into four strips using the width of 0.25 for each one

Find the y coordinate at the end of each strip by reading the value from the graph or substituting the x coordinate into y equals cube root of x

EAG-REWRITTEN-Example-5-3-20-Diagram-2, downloadable IGCSE & GCSE Maths revision notes

Method 1 (summing the areas of several trapeziums)

Find the area of each strip by using the formula for the area of a trapezium

table row cell 1 half open parentheses 0 plus 0.63 close parentheses open parentheses 0.25 close parentheses end cell equals cell 0.07875 end cell row cell 1 half open parentheses 0.63 plus 0.79 close parentheses open parentheses 0.25 close parentheses end cell equals cell 0.1775 end cell row cell 1 half open parentheses 0.79 plus 0.91 close parentheses open parentheses 0.25 close parentheses end cell equals cell 0.2125 end cell row cell 1 half open parentheses 0.91 plus 1 close parentheses open parentheses 0.25 close parentheses end cell equals cell 0.23875 end cell end table

Add up all the areas

0.07875 plus 0.1775 plus 0.2125 plus 0.23875 equals 0.7075

Area almost equal to 0.708

Method 2 (using the trapezium rule)

The trapezium rule is Area equals 1 half open parentheses y subscript 0 plus 2 open parentheses y subscript 1 plus y subscript 2 plus... plus y subscript n minus 1 end subscript close parentheses plus y subscript n close parentheses cross times h

h is the width of each trapezium (along the x axis)

h equals 0.25

y subscript 0 to y subscript n are the y-values (the lengths of the sides of the trapeziums)

table row cell y subscript 0 end cell equals cell 0 space space space y subscript 1 equals 0.63 space space space y subscript 2 equals 0.79 space space space y subscript 3 equals 0.91 space space space y subscript 4 equals 1 end cell end table

Substitute the values into the formula

Area equals 1 half open parentheses 0 plus 2 open parentheses 0.63 plus 0.79 plus 0.91 close parentheses plus 1 close parentheses cross times 0.25

Work this out on your calculator

Area equals 0.7075

Area almost equal to 0.708

(b) Is your answer an overestimate or underestimate? Explain how you know.

Answer:

The top of every trapezium is below the curve, therefore this is an underestimate of the area

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths Content Creator

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.