# Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

Compute the integral describing the area between two plane curves

Contributed by:
Wolfram|Alpha Math Team

ResourceFunction["AreaBetweenCurvesIntegral"][{ returns an Inactive integral representing the area of the enclosed region between the functions |

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 |

Compute an integral representing the area between two curves:

In[1]:= |

Out[1]= |

Plot the region:

In[2]:= |

Out[2]= |

Activate the integral to compute the area:

In[3]:= |

Out[3]= |

Find the area of the region enclosed by two curves:

In[4]:= |

Out[4]= |

In[5]:= |

Out[5]= |

Plot the region in question:

In[6]:= |

Out[6]= |

A region where the curves do not meet:

In[7]:= |

Out[7]= |

In[8]:= |

Out[8]= |

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

In[9]:= |

Out[9]= |

Visualize the regions:

In[10]:= |

Out[10]= |

Find the area between two curves containing parameters:

In[11]:= |

Out[11]= |

The result may be conditioned on parameters:

In[12]:= |

Out[12]= |

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

In[13]:= |

Out[13]= |

In[14]:= |

Out[14]= |

Compute the area of a disk:

In[15]:= |

Out[15]= |

In[16]:= |

Out[16]= |

Visualize the disk:

In[17]:= |

Out[17]= |

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[18]:= |

Compute an integral representing the area between *f* and *g*:

In[19]:= |

Out[19]= |

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

In[20]:= |

Out[20]= |

Though their areas are the same, the regions are very different:

In[21]:= |

Out[21]= |

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

In[22]:= |

Out[22]= |

Activate the integral to compute the area:

In[23]:= |

Out[23]= |

Use the resource function AreaBetweenCurves to compute the area directly:

In[24]:= |

Out[24]= |

- 2.0.0 – 23 March 2023
- 1.0.0 – 11 May 2020

This work is licensed under a Creative Commons Attribution 4.0 International License