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Problem 16 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 = 4 - 2x $ , $ y = 0 $ , $ x = 0 $ ; about $ x = -1 $

Answer

$V=\frac{40 \pi}{3}$

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Video Transcript

we know that the general formula is gonna be vey is two pi times the interval from zero to of radius times height, deal backs. Now we know using the cylindrical show, Matt, that our radius is exports one and our height is four minus two x. So radius times height were simply plugging in times d of X, which means now we have two pi times the intro from 0 to 2 Now use the foil such distributed method to get negative two x squared plus two acts plus four d of X. Now we integrate. What we're gonna be doing is we're gonna be increasing the exponents by one using the power method. And then we're gonna be dividing by the new exponents as you can see what we're doing over here. Now we know our bounds air from zero to It's time to now plug end to our integral negative. Two times two cubed, divided by three plus two squared plus four times to minus zero, which gives us 40 pi divided by three