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In these applications, all area is counted positively. If we slice horizontally and want the unsigned area between two functions from a to b we need to evaluate \displaystyle \int_a^b |f(y)-g(y)| \, dy . If you are using a numerical tool, you can just evaluate this integral using the absolute value of the integrand. In other cases if the curves for the two functions cross in the interval from a to b, it may be easier to divide the interval up into parts where the curves don't cross and evaluate each separately.
Think of each slice as being dy tall and the length is then the difference between f (y) and g(y).
The first example shown is \displaystyle \int_0^1 \sqrt{y}-0 \, dy . Move the y slider to move the sample rectangle.
The second example is \displaystyle \int_0^1 y^2- y \, dy .

Software/Applets used on this page

Webcompmath and calculus applets are free, open-source software aimed at supporting the teaching of mathematics.These applets show various types of graph, that can be altered in some way by clicking on the display or by using sliders provided. Zooming in and out may be achieved as follows:
  • buttons may be provided

  • zoom in by clicking the left mouse button

  • zoom in by dragging a out a box with the left mouse button

  • zoom out by shift-clicking the left mouse button

  • drag the display by clicking and dragging with the right mouse button

WebCompMath is a collection of applets written by Thomas S. Downey and based on the original JCM project by Professor David J. Eck (Hobart and William Smith Colleges, Geneva, New York).

� Copyright 2001 David J. Eck

� Copyright 2007 Thomas S. Downey


absolute value

The modulus, |x|; the number without the sign.


the study of change; a major branch of mathematics that includes the study of limits, derivatives, rates of change, gradient, integrals, area, summation, and infinite series. Historically, it has been referred to as "the calculus of infinitesimals", or "infinitesimal calculus".
There are widespread applications in science, economics, and engineering.


A diagram showing a relationship between two variables.
The diagram shows a vertical y axis and a horizontal x axis.


the anti-derivative


the function to be integrated


A collection of sampling units drawn from a sampling frame.

Full Glossary List

This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AP Calculus AB (USA)4IntegrationDefinite integration-
AP Calculus BC (USA)4IntegrationDefinite integration-
Methods (UK)M9IntegrationDefinite integration-
Scottish Advanced HighersM1IntegrationDefinite integration-
Scottish (Highers + Advanced)AM1IntegrationDefinite integration-
Universal (all site questions)IIntegrationDefinite integration-