Page 64 - Calculus for the Clueless, Calc II

Chapter 11—
Volumes By Rotation and Section

This chapter is the first of two that are based on material from my first book, Calc Helper, the beginnings of this
course. The material in this chapter is in many Calc I courses. However, at my school, it is in Calc II. The
material in the next chapter, "Conic Sections—Circle, Ellipse, Parabola, Hyperbola," is taught in either Calc I or
Calc II, depending on the book and the course. In my school, it is in the non-math-major Calc I and math-major
Calc II. Therefore, this subject deserves to be treated in both books.

The first topic is to find volumes of rotations. This is very visual. If you see the picture, the volume is easy. If
not, this topic is very hard.

Imagine a perfectly formed apple with a line through the middle from top to bottom. We can find the volume
two different ways. One way is by making slices perpendicular to the line (axis). (We will do the other way
later with an onion.) Each slice is a disc, a thin cylinder. Its volume is πr h, where h is very small. If we add up
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all the discs, taking the limits properly, we get the volume.

We will take the same region in six different problems, rotating this region differently six times and getting six
different volumes.

Example 5—

Find the volume if the region R is rotated about the x-axis.

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The volume of each disc is πr h: h = ∆x; r = y. So r = y = x, and x goes from 0 to 9.
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