The volume of a solid with known cross sections can be calculated by taking the definite integral of all the cross sections, with being equal to a single section.

For example, suppose we want to find the volume of the solid with each cross section being a circle, with the diameter of each cross section being the distance between and the x-axis from 0 to 2. Since one cross section will be equal to and , a single cross section will have an area of or . The volume will therefore be equal to

This method can be used to derive geometric formulas. Here, we will find the volume of a pyramid with sides of the base equal to and height . Since , all that is needed is to find a function of to describe the area of a cross section of the pyramid at height . Using trigonometry, we can find that . All that is needed is to isolate so that and square it (since this function describes the area of a cross section), then integrate from to . It is important to remember that and are constants and must be treated as such.

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