Function Repository Resource:

# AreaBetweenCurvesIntegral (1.0.0)current version: 2.0.0 »

Compute the integral describing the area between two plane curves

Contributed by: Wolfram|Alpha Math Team
 ResourceFunction["AreaBetweenCurvesIntegral"][{f,g},{x,xmin,xmax}] returns an Inactive integral representing the area of the enclosed region between the functions f(x) and g(x) over the interval xmin

## Details and Options

ResourceFunction["AreaBetweenCurvesIntegral"] works with real‐valued functions over the Cartesian coordinate system.
The area between f(x) and g(x) is defined as .
When f(x)g(x), the area between the two curves is .
When f(x) and g(x) only meet at x=xmin and x=xmax, the area is taken to be that of the enclosed region.
When f(x) and g(x) do not meet at x=xmin or x=xmax, the boundary of the enclosed region will contain vertical line segments joining the curves.
When f(x) and g(x) intersect for some xmin<x<xmax, the area will be that of multiple enclosed regions.
The following option can be given:
 Assumptions \$Assumptions assumptions on parameters

## Examples

### Basic Examples (3)

Compute an integral representing the area between two curves:

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Plot the region:

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Activate the integral to compute the area:

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### Scope (4)

Find the area of the region enclosed by two curves:

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Plot the region in question:

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A region where the curves do not meet:

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With multiple enclosed regions, the integrand will use Abs to return a positive area:

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Visualize the regions:

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Find the area between two curves containing parameters:

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### Options (2)

#### Assumptions

The result may be conditioned on parameters:

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Make an assumption about the parameter to then evaluate the area:

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### Applications (2)

Compute the area of a disk:

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Visualize the disk:

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Cavalieri's principle states that the area between two curves does not change when each curve is shifted by the same amount. Here are three functions:

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Compute an integral representing the area between f and g:

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This is the same as the integral obtained after shifting both f and g by h:

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Though their areas are the same, the regions are very different:

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### Properties and Relations (3)

Find the integral representing the area between sin(x) and cos(x) over the interval (0,π):

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Activate the integral to compute the area:

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Use the resource function AreaBetweenCurves to compute the area directly:

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## Publisher

Wolfram|Alpha Math Team

## Version History

• 2.0.0 – 23 March 2023
• 1.0.0 – 11 May 2020

## Author Notes

To view underlying source code, evaluate the following:

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