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Problem 14 Medium Difficulty

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ x + y = 4 $ , $ x = y^2 - 4y + 4 $

Answer

$V=\frac{27 \pi}{2}$

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Video Transcript

If you wish to draw the diagram for this question, you will see that it will look something like this giving us a shaded region over here, as you can see, which means that we have height is four minus y minus princes y squared minus four y plus four. Remember, this is because we're looking at outer minus inner. And remember, the radius is simply why so the height is three. Why minus y squared? As I said, the radius is why not? Formula for volume is two pi times are bound in this case or bounder from 03 radius times height. Why times three My minus y squared radius times height. Pretty straightforward if you know what to plug in. Now that we get this, let's clean this up a little bit before we start integrating. We can simplify this by using the distributive property which you London algebra to mean three y squared minus y cubed d y. Now that we're integrating, we know we're using the power method, which means we increase the exponents by one. So in this case, the y squared becomes why cubed? We divide by the new experiment which in this case is actually just gonna be one. Because if we divide by three, but we also have free on the outside three divided by three is just once we actually end up with why cubed? Which means we have why cubed minus y to the fourth over four. And then this is from 0 to 3. Now we can plug in zero. Plugged in simply yields us zero, which means you don't need to worry about that Lower bound. It's just the upper bound to be worried about in this problem here, you should end up with 27 pi, divided by two as your solution.