By Robert Miller

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This is not true, of course, but the examples are extraordinary. Example 7— We will take an infinite area, rotate the region, and get a finite volume!!!!!!!!! Here's our infinite area: Volumes are determined by sections. Much more on this is in Chap. 11 in this book. Amazing!!!!!! Infinite area rotated gives finite volume. This one will totally blow your mind. We will now take a finite region, rotate it, and get an infinite volume, which would seem impossible after the last example. It is not!!!!

4) or (5), we get C = 3. The last part of this long integral chapter is called— Miscellaneous Miscellaneous means anything that doesn't fit into any other part. So all the extra goes into this part, which makes it more miserable for you. Example 24— Sometimes the simplest substitutions work. We let u = x + 1. du = dx. This transfers the power to the monomial and allows us to multiply out the expression (x = u - 1). You sharp-eyed readers will note that there are at least two other ways to do this problem.

6. So the integral is Note I know this last trick is one almost no one will use. But I've left it in because my original editor, David Beckwith, was so great. This is one of his favorite tricks. Thanks, David. Example 11— Find the area of r = 4 cos 6θ. We know we can slide the curve y = f(x) + a units to the right by replacing x by x - a. In the same way, we can rotate r = f(θ) through a counterclockwise angle +α by replacing θ by θ -α. Thus, by rotating our curve by 15º = π/12 radians, r = 4 cos 6(θ - π/12) = 4 cos (6θ - π/2) = 4 sin 6θ, which is exactly the curve in Example 10!!!!!!