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Numerade Educator

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Problem 54 Hard Difficulty

Find the volume of the described solid $ S $.
The base of $ S $ is a circular disk with radius $ r $. Parallel cross sections perpendicular to the base are squares.

Answer

$\frac{16 r^{3}}{3}$

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Video Transcript

in order to find the volume we know we're using the formula from negative art are a of ox d ox formula specified in the textbook. Which means we know we have your order. A of axes are in. Max is equivalent to the square root of R squared, minus X squared. Plus it's word of R squared minus X squared. And this whole thing is squared. Now, you should have learned the foil method, which means we can actually expand this. When you square square root, it just cancels off all the stuff on the outside and what's left is on the inside. So in other words, this could be written as four times r squared minus X squared. Okay, this is our backs. Which means we can now plug in again. Four can be pulled out. We've learned this before, in previous chapters that Constance can be pulled out. Okay, Now, we've also learned that we can actually split up into girls. We have two parts. We can split up into girls and write this with separate notations. Now, given this we know we cannot integrate, which means we can use the power rule, increase the exponent by one divide by the new exponents. We have X cubed over three instead of X squared, for example, substitute in our bounds. That's the next stop, and we end up with 16 over three are cute.