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Problem 15 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 = x^3 $ , $ y = 8 $ , $ x = 0 $ ; about $ x = 3 $

Answer

$V=\frac{264 \pi}{5}$

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Video Transcript

If you wish to draw this diagram, if it will help you, then you can draw it just so you can see the shaded region like this. You can see that we're gonna be cutting it this way like this. So we'll draw the cut and red sea you can see and you can see are shaded region and you can see like this. Okay, now that we have this, we know radius is three minus acts and the height is eight minus x cubed. Remember BA formula? The general formula for volume is two pi times the integral from our bounds, which in this case is 0 to 2 radius times height. So we have radius three months X times height, eight minus x cubed. And then remember, we need d of X on the outside. If it was why it would be d of what? Before we integrate this that's actually distribute it to make it easier to read. It's easier to integrate when we have a lot of separate terms than parentheses. Parentheses make it a lot more difficult to integrate. Okay, Now that we have this, we know we can now use the integration method which means through the power method. We increased the exponents by one and we divide by the new exponents, which means now it's time to plug End to the fifth over. Five minus 3/4 times two to fourth minus four times two squared, plus 24 times to minus zero. The minus here doesn't affect this problem. We now have 264 pi divided by five.