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

$ y = x^2 $ , $ y = 6x - 2x^2 $

8$\pi$

Applications of Integration

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in order to use the method of cylindrical shells. The first thing we need to do is find intersection points. In other words, set our two values X squared and six x minus two X squared off of X and G of X, equal to each other. So for acts to better intersection points which will then become our bounds, says you can see this problem requires that we figure out our own bounds. Now, remember, the formula for volume is two pi times bounds from A to B integral or Veitch, which means radius times, height of D of X. So now, given this, we can actually planning what we know, as we just specified are too. Intersection points were zero in two. Therefore, lowest one goes in the bottom. Hi, Simon goes in the top acts time 66 months, three x squared again Our times h times d backs. Okay, pull out our constance three times to a sex sex pies pulled out on the outside. Use the power rule. Increase the exponents by one divide by the new exponents were the point now where we can actually plug in our bounds. You can probably tell from looking at this, that if we plug in zero, we literally just end up with zero. So you don't have to write that as long as you recognize the fact that the zero doesn't affect the problem in this context, simplify this further. We end up with eight pi.