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Problem 37 Hard Difficulty

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ y = -x^2 + 6x - 8 $ , $ y = 0 $ ; about the y-axis


$V=8 \pi$

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Video Transcript

we know the radius is gonna be ex. The height of the cylindrical shell is negative. X squared plus X X minus eight. As specified in the problem. Let us plug into the formula for V. We have two pi times the integral from 2 to 4. Those are bounds times radius. We just at his axe times height, which has said his night of X squared plus six acts minus eight times D of ax. Okay, we're gonna be integrated when we integrate. Use the power rule, which means we increased exported by one. And then we divide by the new exponents. So remember, this was negative. X squared. When we multiplied this by acts, we end up with negative acts. Cute, which means negative X Cube becomes night of X to the fourth over four. So remember that we have to be distributing by what's on the outside. For all three of these terms, there's an algebraic concepts you probably learned years ago. Okay, now that we've got our integral, we know it's time to plug in. Now, this is a little bit long, but you basically just have to plug in your upper bound plugged in minus your lower bound plugged end minus two pod times negative to the fork over four plus two times two cubed minus four times two squared, which gives us eight pi as our solution.