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Numerade Educator

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Problem 59 Hard Difficulty

Find the volume of the described solid $ S $.
The base of $ S $ is the same base as in Exercise 58, but cross-sections perpendicular to the x-axis are isosceles triangles with height equal to the base.

Answer

$\frac{8}{15}$ units $^{3}$

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Video Transcript

Okay, we know we've been given one minus X squared. We know this itself has to be squared because this is only half of what we need, which means that we are done given one minus two X squared plus acts to the fourth. And then remember that on top of this, the original formula had 1/2 on the outside because it was 1/2 times off of X squared. Okay, now that we have this, we know we can write our integral from 0 to 1. And then remember, we know it is symmetric. Therefore, we can multiply by two. This is a crucial step. People often forget that. But remember, either you can write to integral or you could just double it. In this case, I think it's probably easier to just double it now, integrate using the power rule, which means increased exploded by one divide by the new exponents. And then when we plug ins here, we just end up with zero. So you don't even have to write. Plus zero. We end up with eight over 15 units cubed