Function Repository Resource:

# AreaBetweenCurves (5.3.1)current version: 5.3.2 »

Find the area between two plane curves

Contributed by: Wolfram|Alpha Math Team
 ResourceFunction["AreaBetweenCurves"][{f,g},{x,xmin,xmax}] finds the area of the enclosed region between the functions f(x) and g(x) over the interval xmin < x < xmax.

## Details

ResourceFunction["AreaBetweenCurves"] 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 (1)

Find the area between two curves:

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### Scope (4)

Find the area of the region enclosed by two curves:

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Where the curves do not meet:

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With multiple enclosed regions:

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Between curves containing parameters:

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### Generalizations and Extensions (3)

Find the area over an unbounded interval:

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Curves with discontinuities over intervals:

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With singularities:

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### Options (2)

#### Assumptions

The result may be conditioned on parameters:

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Make an assumption about the parameter:

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### Applications (3)

Compute the area of a 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:

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The population of a region is currently growing at a rate of 35.208 ⅇ0.0083 t hundred people per year. It is thought that a large spike in employment opportunities can drop the growth rate to 24.098 ⅇ0.0071 t hundred people per year over the next five years. Find how many fewer people will be born if such a spike occurs:

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### Properties and Relations (5)

Area is always non-negative:

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The order in which the curves are specified does not matter:

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Find the area of multiple enclosed regions:

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Sum over each enclosed region instead:

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The area between two curves is the integral of the absolute value of their difference:

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### Possible Issues (2)

The integral defining the area between two curves may not converge:

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In such cases, AreaBetweenCurves throws a message:

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Functions must be real-valued over the entire range of integration. Here is imaginary for x>1:

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AreaBetweenCurves throws a message to warn the user:

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Restricting the domain of integration yields a result:

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## Publisher

Wolfram|Alpha Math Team

## Version History

• 5.3.2 – 22 March 2023
• 5.3.1 – 22 March 2023
• 5.3.0 – 12 May 2021