To calculate the area between curves take the integral of over the region you want to find.

Example of the area between the curves and between and .

Single Enclosed Area

To find the area enclosed between two curves first find the points which they cross, then find which function is above and which is below and then perform an integral of the upper function minus the lower function from the left endpoint to the right endpoint.

Example

Compute the finite area between the curves and .

Finding the endpoints

Visually we can see that is the upper function and is the lower function.

Area between Curves that "Crossover"

To find the area between curves with crossovers (ie. locations where they alternate between above and below), first find the locations where the intersect and then determine in each of these sections which is the higher and lower function, then repeat the process of finding a single enclosed area for each of these "pockets".

Examples

Example 1

Find the area between the curves and between and .

Finding the intersection points

Example 2

Find the finite area between the curves and .

Finding the endpoints

Taking the upper branch

Taking the lower branch

Using these endpoints we need to divide the area into 2 sections