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Numerade Educator

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Problem 25 Hard Difficulty

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_2 $ about $ AB $

Answer

$\frac{13}{45} \pi$

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Video Transcript

uh for giving a figure which consume the region and you're lying and rest defying the volume generated by rotating it. This region about this line, the region is are too. And we're lab rotating it about the line. Baby man looking at the figure where he's just like, well you have the upper bound on the region. Why equals 12 bound on the side on the right side. Should say is X equals uh Why he goes the 4th root of X. And you know, bounds. No, it's not. The bound on the left hand side is just X equals zero. And if we're rotating about a B, This is the same as rotating about the line x equals one. Yeah, he has a pit bull to and with human arm. Now looking at a figure, it's clear that we will get a washer type solid that we use the washer method About 100 lb. So we need to find the radius radi I of the washer. Yeah, we're looking at the figure. It appears that the outer radius of the washer with one and the inner radius to sort of is X equals one minus X equals y. Before and so using the washer method, the volume is equal to high times integral. From y equals zero to y equals one of the outer radius, one squared minus the inner radius one minus y to the fourth squared Dy They get to a certain age where they're like, well I guess I'm smart now. I guess I've smart years old. Yeah. And this is a pretty easy integral to evaluate. Once you do, you should get 13 pie. Talk to the judge all Over 45 10 guys sit there please.