# 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:

 In:= Out= Plot the region:

 In:= Out= Activate the integral to compute the area:

 In:= Out= ### Scope (4)

Find the area of the region enclosed by two curves:

 In:= Out= In:= Out= Plot the region in question:

 In:= Out= A region where the curves do not meet:

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

 In:= Out= Visualize the regions:

 In:= Out= Find the area between two curves containing parameters:

 In:= Out= ### Options (2)

#### Assumptions

The result may be conditioned on parameters:

 In:= Out= Make an assumption about the parameter to then evaluate the area:

 In:= Out= In:= Out= ### Applications (2)

Compute the area of a disk:

 In:= Out= In:= Out= Visualize the disk:

 In:= Out= 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:

 In:= Compute an integral representing the area between f and g:

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

 In:= Out= Though their areas are the same, the regions are very different:

 In:= Out= ### Properties and Relations (3)

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

 In:= Out= Activate the integral to compute the area:

 In:= Out= Use the resource function AreaBetweenCurves to compute the area directly:

 In:= Out= ## 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:

 In:= 