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Problem 17 Medium Difficulty

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$ y = 4x - x^2 $ , $ y = 3 $ ; about $ x = 1 $

Answer

$\frac{8}{3} \pi$

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Video Transcript

The method of cylindrical shells implies the radius is X minus one, and the height is four acts minus X squared minus three. This information was also given in the problem. Now we know the formula for an intern role for volume is two pi times the intervals from the bound, which is in this case from 1 to 3 times radius times height, which we're gonna write as X minus one times four acts minus X squared minus three D X. Now, before we integrate this, I would highly recommend simple, fine ease into separate terms. Parentheses don't work well with integration. It gets really messy and confusing. So if you do what I'm doing using the floor mat that you learned from algebra, you simplify them. You can now integrate using the power Mathis increasing the expert by one dividing by than you exponents do this for each of the terms. You can see this a lot neater than having to deal with parentheses. Bounds air from 1 to 3. It's now time to plug in. We're plugging in are bound of three. First it's gonna be the bound of three plugged in minus the back of one plug day. Remember, it's always top minus boredom, so top minus bottom 5/3 minus one fork, minus seven over two plus three gives US eight pi divided by three