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The Second Theorem of Pappus is in the same spirit as Pappus's Theorem on page 565, but for surface area rather than volume: Let $ C $ be a curve that lies entirely on one side of a line $ l $ in the plane. If $ C $ is rotated about $ l $, then the area of the resulting surface is the product of the arc length of $ C $ and the distance traveled by the centroid of $ C $ (see Exercise 47).

(a) Prove the Second Theorem of Pappus for the case where $ C $ is given by $ y = f(x), f(x) \ge 0 $ and $ C $. is rotated about the x-axis.

(b) Use the Second Theorem of Pappus to compute the surface area of the half-sphere obtained by rotating the curve from Exercise 47 about the x-axis. Does your answer agree with the one given by geometric formulas?

(a) 2$\pi \overline{y}$ is the total distance the centroid traveled around the $x$ axis

so $S=2 \pi \overline{y} L=d L$

(b) The surface area of a half sphere is $S_{a}=\left(4 \pi r^{2}\right) / 2$

$S_{a}=2 \pi\left(4^{2}\right)=32 \pi$

We have confirmed the geometric formula.

Applications of Integration

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