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Problem 31 Medium Difficulty

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ y = e^{-x^2} $ , $ y = 0 $ , $ x = -1 $ , $ x = 1 $

(a) About the x-axis
(b) About $ y = -1 $

Answer

a) $V \approx 3.75825$
b) $V=13.14$

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Deema M.

February 16, 2021

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 1/ 16 x2, x = 5, y = 0; about the y-axis

Video Transcript

problem asks set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral, correct if I decimal places and our equation is Y equals eat the negative X squared, Y equals E to the power of negative X squared. And our region is Y equals zero X equals negative one and X equals one. So part A asks what happens when you set it up around the excel about the X axis. So for here for this problem we have that the volume is equal to the integral from negative 1 to 1. Where these values come from our X values and then we have pi R squared D R and r R. Here is actually the equation itself. So we have our equals each of the eight to the power of negative X squared. So we have high times E to the power of a negative x squared squared dx. And if we compute this using an online computer, we get that this is approximately equal to 3.758 to 3 for part B. It's a little more interesting. It asks what happens about why equals negative one. So for here the balance are exactly the same. We have negative 1 to 1 pie. And then what is different is the term inside the parentheses. So we have E to the negative X squared minus negative one where this negative one comes from, that's negative one here and then we square everything and then minus one squared where this one square comes from a whole of radius one. And so if you compute this integral, we get that. This is approximately equal to 13.14305