Function Repository Resource:

AreaBetweenCurvesIntegral

Compute the integral describing the area between two plane curves

Contributed by: Wolfram|Alpha Math Team

ResourceFunction["AreaBetweenCurvesIntegral"][{f,g},{x,x_min,x_max}]

returns an Inactive integral representing the area of the enclosed region between the functions f(x) and g(x) over the interval x_min<x<x_max.

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=x_min and x=x_max, the area is taken to be that of the enclosed region.

When f(x) and g(x) do not meet at x=x_min or x=x_max, the boundary of the enclosed region will contain vertical line segments joining the curves.

When f(x) and g(x) intersect for some x_min<x<x_max, the area will be that of multiple enclosed regions.

The following option can be given:

Assumptions

$Assumptions

assumptions on parameters

Examples

Basic Examples (2)

Compute an integral representing the area between two curves:

In[1]:=

$area = ResourceFunction[ "AreaBetweenCurvesIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{x, x^2}, {x, 0, 1}]$

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In[2]:=

$Plot[{x, x^2}, {x, 0, 1}, AspectRatio -> Automatic, Filling -> {1 -> {2}}]$

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Activate the integral to compute the area:

In[3]:=

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Scope (4)

Find the area of the region enclosed by two curves:

In[4]:=

$area = ResourceFunction[ "AreaBetweenCurvesIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{x^2 - 1, 1 - x^2}, {x, -1, 1}]$

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In[5]:=

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In[6]:=

$Plot[{x^2 - 1, 1 - x^2}, {x, -1, 1}, AspectRatio -> Automatic, Filling -> {1 -> {2}}]$

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Where the curves do not meet:

In[7]:=

$ResourceFunction[ "AreaBetweenCurvesIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{Sin[2 x]/2 + 2, Cos[x]/2 - 1}, {x, 0, 2 \[Pi]}]$

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In[8]:=

$Plot[{Sin[2 x]/2 + 2, Cos[x]/2 - 1}, {x, 0, 2 \[Pi]}, AspectRatio -> Automatic, Filling -> {1 -> {2}}]$

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

In[9]:=

$ResourceFunction[ "AreaBetweenCurvesIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{x/2, x^2}, {x, -1/2, 1/2}]$

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In[10]:=

$Plot[{x/2, x^2}, {x, -1/2, 1/2}, AspectRatio -> Automatic, Filling -> {1 -> {2}}]$

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Between curves containing parameters:

In[11]:=

$ResourceFunction[ "AreaBetweenCurvesIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{a x, x^3}, {x, 0, 1}]$

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Options (2)

Assumptions

The result may be conditioned on parameters:

In[12]:=

$ResourceFunction[ "AreaBetweenCurvesIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{Sqrt[a x^2], x}, {x, -1, 1}]$

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

In[13]:=

$area = ResourceFunction[ "AreaBetweenCurvesIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{Sqrt[a x^2], x}, {x, -1, 1}, Assumptions -> a > 1]$

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In[14]:=

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Applications (2)

Compute the area of a disk:

In[15]:=

$area = ResourceFunction[ "AreaBetweenCurvesIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{Sqrt[1 - x^2], -Sqrt[1 - x^2]}, {x, -1, 1}]$