# w07 - 1/x(a Compute the volume of the “trumpet” formed...

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Math 152 Workshop 7 Spring 2009 1. The curve y = e - x , x 0, is revolved about the x -axis. Does the resulting surface have ﬁnite or inﬁnite area? (Remember that you can sometimes decide whether an improper integral converges without calculating it exactly.) 2. Suppose R is the region in the plane enclosed by y = x 2 and y = 4. (a) Compute the perimeter P and area A of R , and then compute the ratio Q = A/P 2 . Note By squaring the perimeter the ratio becomes independent of the units chosen to measure the region. (b) Compute this ratio Q = A/P 2 for these four regions: the region R , a square, a circle, and an equilateral triangle. Draw the ﬁgures in increasing order of Q . 3. Torricelli’s Trumpet: Consider the unbounded region on the xy - plane deﬁned by R : x 0 , 0 y
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Unformatted text preview: 1 /x . (a) Compute the volume of the “trumpet” formed by rotating the region R about the x-axis. (b) Prove that the surface area of the trumpet is inﬁnite. Remember that to prove this you don’t need to ﬁnd the antiderivative of the resulting integrand, but you can compare this integrand with one with easier antiderivative, and divergent integral. (c) Discuss the apparent counterintuitive results obtained: According to (a), you can ﬁll the trumpet with a ﬁnite amount of paint, but according to (b), apparently you need an inﬁnite amount of paint to coat the inside of the trumpet. Note: there is an interesting article about this on Wikipedia....
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## This note was uploaded on 03/24/2011 for the course CALCULUS 152 taught by Professor Sosa during the Spring '11 term at Rutgers.

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