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Problem 12 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 = -3y^2 + 12y - 9 $ , $ x = 0 $

Answer

$V=16 \pi$

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Video Transcript

for this question, It may be helpful to draw a diagram. We can see our heightened our with and we can see are shaded region like this on We can see that the radius is gonna be why and the height is gonna be negative. Three y scored post 12 Why minus nine formula for volume Just to refresh your memory to pi times the intro from our bounds. In this case, it's 123 times radius times height. Now plug it Radius is why height is negative. Three y squared plus 12. Why minus nine And then we know this is gonna be multiplied by D. Why this gives us to pile on the outside Thames, the integral from 1 to 3. What's simple, finest negative Three. Why cubed plus 12 y squared minus nine. Why do you why, Okay, now that we integrate, we know we integrate by using the power rule, which means we increase the expert by ones and this gets increase it before divide by the new exponents. We do that for each of the three terms in this context, keeping in mind that our constant constituents down the outside now it's important that we plug in. We know we have two bounds to plug in. It's always top bound, minus the bottom bound when we are doing into role. So if this is the top bound, we are subtracting the bottom bounce again. It's the same constant on the outside that stays the same for both of them, and then we're playing in the two values. This is equivalent to 16 pie, which is R V.