# Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

Find the area between two plane curves

Contributed by:
Wolfram|Alpha Math Team

ResourceFunction["AreaBetweenCurves"][{ finds the area of the enclosed region between the functions |

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 |

Find the area between two curves:

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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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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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The result may be conditioned on parameters:

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Make an assumption about the parameter:

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Compute the area of a disk:

In[18]:= |

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

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In such cases, AreaBetweenCurves throws a message:

In[35]:= |

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

In[36]:= |

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AreaBetweenCurves throws a message to warn the user:

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Restricting the domain of integration yields a result:

In[39]:= |

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

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