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For the following exercises, use the theorem of Pappus to determine the volume of the shape.A general cone created by rotating a triangle with vertices $(0,0), \quad(a, 0),$ and $(0, b)$ around the $y$ -axis. Does your answer agree with the volume of a cone?

$\frac{a^{2} b \pi}{3}$

Calculus 2 / BC

Chapter 6

Applications of Integration

Section 6

Moments and Centers of Mass

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Use the Theorem of Pappus …

Okay, so we have a triangular shape around the Y axis. So let's say we have something like this and we're asked to rotate it. Yeah, I saw Area is equal to 1/2 our length over here. It's got us a and our hides over here. Let's call to speak. Okay, so that's 1/2 a times B. It becomes this a b over to tell if a triangle hazardous ease from this point here just once here in this place yet actually, we only looked from Joe to these Deep wants. Our center of mass is the sum of our X points, which is Darryl plus a Let's start all over three and some of our white points, which is open. So being over three, right, because we have our verte sees at socom a job from a comma Joe and become a job. So this becomes a over three comma, be over three, and now we need to find our distance. But our distance is this equal to two high times our export, which is a over three. So that's just to a pie over three. And based on the dam of purpose, a Times T is equal to our volume, which is equal to a B over three over two times to a pie over three, which becomes a squared B by over three.

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