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Cavalieri's PrincipleThis section explains a second way to see the relation between
double integrals and interated integrals. (You may or may not need to learn
Cavalieri's Principle  check with your instructor.) An alternate approach to finding volumes (and hence double
integrals)  the socalled Slice Method
 was formulated by Cavalieri and is expressed mathematically in
We already used this idea to compute volumes of revolution. Suppose $W$ is created by rotating the graph of $y = f(x),\, a \le x \le b,$ about the $x$axis. When $P_x$ is a plane perpendicular to the $x$axis, then the slice of $W$ cut by $P_x$ is a disk of radius $f(x)$. Here $A(x) = \pi f(x)^2$, so we recover the familiar result $$\hbox{ volume of} \ W \ = \ \pi \int_a^b\, f(x)^2\, dx$$ for a volume of revolution. But Cavalieri's Principle does not require the crosssections to be triangles or disks, as we will see. We consider an example that we already computed as an iterated integral.
