# Finding the Volume of a Solid with Similar Cross Sections

Finding the volume of a prism or cylinder is pretty straightforward. But what if you need to find the volume of a shape like the one shown here? In this case, slicing parallel to the base always results in the same shape — a circle — but the area may differ. That is, the solid has *similar *cross sections rather than congruent ones.

You can estimate this volume by slicing the solid into numerous cylinders, finding the volume of each cylinder by using the formula for constant-height solids, and adding these separate volumes. Of course, making more slices improves your estimate. And, as you may already suspect, taking the limit of a sum of these slices as the number of slices increases without bound gives you the exact volume of the solid.

Hmmm . . . this is beginning to sound like a job for calculus. Specifically, you can use the meat-slicer method, which works well for measuring solids that have similar cross sections.

Here’s the plan:

Find an expression that represents the area of a random cross section of the solid in terms of

*x**.*Use this expression to build a definite integral (in terms of

*dx*) that represents the volume of the solid.Evaluate this integral.

When a problem asks you to find the volume of a solid, look at the picture of this solid and figure out how to slice it up so that all the cross sections are similar. This is a good first step in understanding the problem so that you can solve it.