AP®MathsCollege BoardCalculus BCStudy GuidesUnit 8: Applications of IntegrationAreas & Arc LengthsArea Between a Curve & x-Axis

Area Between a Curve & x-Axis (College Board AP® Calculus BC) : Study Guide

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

College Board Calculus BCCollege Board Calculus AB

Area between a curve & x-axis

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

  • The value found when calculating the definite integral of a function y equals f open parentheses x close parentheses with respect to x between x equals a and x equals b, integral subscript a superscript b f open parentheses x close parentheses space italic d x

    • as long as f open parentheses x close parentheses greater or equal than 0 on the interval open square brackets a comma space b close square brackets

    • is equal to the area between the curve and the x-axis between x equals a and x equals b

Graph of y = f(x) showing the area under the curve between x = a and x = b, shaded in purple and labeled R; integrals are used to find the area.

  • Consider finding the area between the graph of y equals 5 plus 2 x minus x squared and the x-axis, between x equals 1 and x equals 3

Graph of the function y = 5 + 2x - x^2 with a shaded region between x = 1 and x = 3, area calculated as 28/3 square units using integration.

  • This method of finding areas uses the idea of a definite integral as calculating an accumulation of change

    • f open parentheses x close parentheses times increment x is the area of a rectangle with height f open parentheses x close parentheses and width increment x

    • f open parentheses x close parentheses space d x is the limit of this area element as increment x rightwards arrow 0

    • The integral integral subscript a superscript b f open parentheses x close parentheses space d x sums up all these infinitesimal area elements between x equals a and x equals b

What if I am not told the limits?

  •  If limits are not provided they will often be the x-axis intercepts

    • Set y equals 0 and solve the equation to find the x-axis intercepts first

Graph showing the shaded area R under the curve y = x(5 - x). Finding the x-intercepts first, which are 0 and 5. Calculation shows area of R = 125/6 square units.

  • Remember that the y-axis (i.e. x equals 0) may also be one of the limits

When is the area integral negative?

  •  If the area lies underneath the x-axis the value of the definite integral will be negative

    • However, an area cannot be negative

    • The area is equal to the modulus (absolute value) of the definite integral

  • If the area has some parts which are above the x-axis, and some which are below the x-axis

    • then see the method outlined in the 'Multiple Areas' study guide

Examiner Tips and Tricks

Always check whether you need to find the value of an integral, or an area.

  • When areas below the x-axis are involved, these will be two different values.

Graph of y = x^2 - 6x + 5 with shaded region R between x = 2 and x = 4. Integral calculation shows integral is -22/3, so the area is +22/3

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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.

Dan Finlay

Reviewer: 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.

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