Function Repository Resource:

AreaBetweenCurves

Source Notebook

Find the area between two plane curves

Contributed by: Wolfram|Alpha Math Team

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

finds the area of the enclosed region between the functions f(x) and g(x) over the interval x_min < x < x_max.

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=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 (1)

Find the area between two curves:

In[1]:=

$ResourceFunction["AreaBetweenCurves"][{x, x^2}, {x, 0, 1}]$

Out[1]=

In[2]:=

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

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Scope (4)

Find the area of the region enclosed by two curves:

In[3]:=

$ResourceFunction["AreaBetweenCurves"][{x^2 - 1, 1 - x^2}, {x, -1, 1}]$

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In[4]:=

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

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Where the curves do not meet:

In[5]:=

$ResourceFunction[ "AreaBetweenCurves"][{Sin[2 x]/2 + 2, Cos[x]/2 - 1}, {x, 0, 2 \[Pi]}]$

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In[6]:=

$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:

In[7]:=

$ResourceFunction["AreaBetweenCurves"][{x/2, x^2}, {x, -1/2, 1/2}]$

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In[8]:=

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

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Between curves containing parameters:

In[9]:=

$ResourceFunction["AreaBetweenCurves"][{a x, x^3}, {x, 0, 1}]$

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Generalizations and Extensions (3)

Find the area over an unbounded interval:

In[10]:=

$ResourceFunction[ "AreaBetweenCurves"][{1/(x^2 + 2 x + 2), -1/(x + 2)^2}, {x, 1, \[Infinity]}]$

Out[10]=

In[11]:=

$Plot[{1/(x^2 + 2 x + 2), -1/(x + 2)^2}, {x, 1, 6}, Filling -> {1 -> {2}}]$

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Curves with discontinuities over intervals:

In[12]:=

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With singularities:

In[14]:=

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In[15]:=

$Plot[{1/Sqrt[Abs[x]], x^2}, {x, -1, 1}, Filling -> {1 -> {2}}]$

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Options (2)

Assumptions

The result may be conditioned on parameters:

In[16]:=

$ResourceFunction["AreaBetweenCurves"][{Sqrt[a x^2], x}, {x, -1, 1}]$

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Make an assumption about the parameter:

In[17]:=

$ResourceFunction["AreaBetweenCurves"][{Sqrt[a x^2], x}, {x, -1, 1}, Assumptions -> a > 1]$

Out[17]=

Applications (3)

Compute the area of a disk:

In[18]:=

$ResourceFunction[ "AreaBetweenCurves"][{Sqrt[1 - x^2], -Sqrt[1 - x^2]}, {x, -1, 1}]$

Out[18]=

In[19]:=

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

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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:

In[20]:=

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The population of a region is currently growing at a rate of 35.208 ⅇ^{0.0083 t} hundred people per year. It is thought that a large spike in employment opportunities can drop the growth rate to 24.098 ⅇ^{0.0071 t} hundred people per year over the next five years. Find how many fewer people will be born if such a spike occurs:

In[25]:=

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Properties and Relations (5)

Area is always non-negative:

In[26]:=

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

In[27]:=

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Find the area of multiple enclosed regions:

In[28]:=

$ResourceFunction[ "AreaBetweenCurves"][{Sin[x], Cos[x]}, {x, \[Pi]/4, 5 \[Pi]/4}]$

Out[28]=

Sum over each enclosed region instead:

In[29]:=

$ResourceFunction[ "AreaBetweenCurves"][{Sin[x], Cos[x]}, {x, \[Pi]/4, 3 \[Pi]/4}] + ResourceFunction[ "AreaBetweenCurves"][{Sin[x], Cos[x]}, {x, 3 \[Pi]/4, 5 \[Pi]/4}]$