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Problem 43 Hard Difficulty

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ x = (y - 1)^2 $ , $ x - y = 1 $ ; about $ x = -1 $

Answer

$\frac{117}{5} \pi$

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Video Transcript

Okay. We know the inner radius is why minus one squared plus one, which we know we can, right? Simplified as y squared minus two wide plus two Outer radius is gonna be Y plus one plus one with his wife prostitute. Therefore, we now Harvey is pie times The interval from 0 to 3 off. Outer radius squared minus injurious work. Remember Audi Radius squared Y plus two is outer. It's squared minus inner y squared minus two. Why plus two squared D y. Okay, this simplifies to V is pi times the integral from 0 to 3 negative wide of the fourth plus four. Why cubed? Minus seven y squared plus 12. Why D Y? Which gives us v as pi times negative wide of the fifth over five plus y to the fourth minus seven over three. Why cubed plus six y squared from 0 to 3. Now we can plug in. We know plugging in zero simply yields us zero. So in this context, we actually only have to plug end our upper bound, which is three. Because that is gonna yield us are tangible answer which is gonna be 117. Pi divided by five