The Cross Section Method

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With the disc method, you can find volume of a solid having a circular cross section whose area is A = pi R². This method can be generalized to solids of any shape, as long as you know a formula for the area of an arbitrary cross section. Some common cross sections are squares, rectangles, triangles, semicircles, and trapezoids.

The Cross Section Method
For cross sections of area A(x) taken perpendicular to the x-axis,
                  _b
Volume = §  A(x) dx which is seen below.
                  ^a


For cross sections of area A(y) dy taken perpendicular to the y-axis,
                  _d
Volume = §  A(y) dy which is seen below.
                  ^c



Click here for an example problem.