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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

we know that in order to find the volume were integrating on the access parallel to the axis of rotation, you know, the axis of rotation is X equals one There for the outer radius is gonna be distance from X equals ear to the axis of rotation, which is one minus dear. We're just simply one. And then we know the inner radius is gonna be the distance from why did the fourth equals X to the axis of rotation? X equals one. So again, using the same reasons. Which means we have one minus y to the fourth for the inner now, or volume is pi times integral from 01 of the inner. Okay, so what we just did is we reason the formula pi times outer minus inner and outer and inner. Both individuals scored so one squared is one. This is also squared. So we foiled us. So we So we took this which is the inner. And then we spend them when you square it. Then you use the foil method, right? So now we can simplify before we find the integral. We have to simplify this to go to y to the fourth minus wide of the eighth. Do you? Why now we can integrate. Use the power method, which means increased the experiment by one and divide by the new exponents. We have zero for the second bound of when we plug ins. Here we get zero, which means our solution is 13 point over 45.