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 innite. Remember that to prove this you dont need to nd 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 ll the trumpet with a nite amount of paint, but according to (b), apparently you need an innite amount of paint to coat the inside of the trumpet. Note: there is an interesting article about this on Wikipedia....
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