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Numerade Educator



Problem 48 Hard Difficulty

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.


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Video Transcript

in this problem In part A. We know that there's a currency which we represent, Um, with a function f effects and we irritating this cur dysfunction why they're on the X axis. And we would guys, then such a shape. And we're interested in the surface area as office shamed for my rotating in Toronto the X axis. So from problem 47 we know that why bar is equal to one over l travel from A to B the limits office in struggle for facts multiplied by skirted a four plus DF dx craig which is equal to D. Yes, okay. We also know that s, um is equal to internal form A to B two ply times. Well, now to keep one is this Since we're rotating this around the X axis and less at this object has a central right here since we're returning it around, the XX is we're only interested in distance between the access that we're rotating it at and central. So we're only ingested in white bar. That's why we only had is from elation for white bar. All right, so the business troubled by the sentry in wide direction would be to pi times while screwed one plus um de y r d x squared d s now to buy just a constant so we can rent this one. That's two pi internal from a to B Why scourge route one plus d Y t x squared the eggs And what do we know? We know that why is equal to f off acts. So it means that this sport actually is nothing bad. Why bar So we can then right as as to pine times Why prime? Sorry y bar. However, as you can see, if you compare this inter go with this red one we see that those parts are the same former We have white bar and we have l in division form so we can multiply both sides by el inside Out s is then equal to two pi times Why bar times? Oh, that was part of a So we found an expression for surface area as a function off. Why central it so in problem 47 again we pound Why bar off 1/4 circle to be eight over pie sort of surface area would then be to pi times. Why bar times out gotta be two pi times Y wise ain't will reply. What is L l is the circumference off 1/4 of a circle, so that will be 25 times. Our radius is four divided by four fours. Will cast law pies will cancel our We don't find surface area as 32 Well.