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Numerade Educator

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Problem 21 Medium Difficulty

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ y = xe^{-x} $ , $ y = 0 $ , $ x = 2 $ ; about the y-axis

Answer

a) $V=2 \pi \int_{0}^{2} x^{2} e^{-x} d x$
b) $V=4-\frac{20}{e^{2}} \approx 4.06300$

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Video Transcript

okay part. We know we have radius acts, and height exceeds the night of X. In this context, we know we can use the washer method as a potential option, which means reason the formula to pied times into go from 0 to 2 of radius times height. We just said radius is acts and height is x e to the negative X. This gives us a simplified integral of two pi times the integral from 0 to 2 x squared each the negative acts D x. Now I'm gonna give you a second. Pull out your calculator cause part B says to use your calculator. So if you plug this into any calculator t 84 plus, for example, you end up with 4.6300 as your solution and this is equivalent to four minus 20 over e squared.