Function Repository Resource:

# AreaBetweenCurves

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:

 In:= Out= In:= Out= ### Scope (4)

Find the area of the region enclosed by two curves:

 In:= Out= In:= Out= Where the curves do not meet:

 In:= Out= In:= Out= With multiple enclosed regions:

 In:= Out= In:= Out= Between curves containing parameters:

 In:= Out= ### Generalizations and Extensions (3)

Find the area over an unbounded interval:

 In:= Out= In:= Out= Curves with discontinuities over intervals:

 In:= Out= In:= Out= With singularities:

 In:= Out= In:= Out= ### Options (2)

#### Assumptions

The result may be conditioned on parameters:

 In:= Out= Make an assumption about the parameter:

 In:= Out= ### Applications (3)

Compute the area of a disk:

 In:= Out= In:= Out= Cavalieri's principle states that the area between two curves does not change when each curve is shifted by the same amount:

 In:= In:= In:= Out= In:= Out= In:= Out= 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:

 In:= Out= ### Properties and Relations (5)

Area is always non-negative:

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

 In:= Out= Find the area of multiple enclosed regions:

 In:= Out= Sum over each enclosed region instead:

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

 In:= Out= In:= Out= ### Possible Issues (2)

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

 In:= Out= In such cases, AreaBetweenCurves throws a message:

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

 In:= Out= AreaBetweenCurves throws a message to warn the user:

 In:=   Out= Restricting the domain of integration yields a result:

 In:= Out= ## 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