A8 - MATH 222 HOMEWORK 8 DUE NOVEMBER 14, 2011 1. P...

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Unformatted text preview: MATH 222 HOMEWORK 8 DUE NOVEMBER 14, 2011 1. P ROBLEMS Problem 1.1. Find the area of the region bounded by the curves y = x2 + 1 and y = 9 − x2 . Problem 1.2. Find the volume of the solid bounded by the surfaces z = x2 + 2y 2 and z = 12 − 2x2 − y 2 . Problem 1.3. Evaluate the integral 1 1 0 y 1 dxdy 1 + x4 Problem 1.4. Evaluate the integral by converting to polar coordinates ∞ ∞ dx dy . (1 + x2 + y 2 )3/2 0 0 Problem 1.5. Find the mass and center of mass of the region bounded by the cardioid r = 1 + cos(θ), assuming the density function is given by ρ = r. Problem 1.6. Suppose that a square hole with sides of length 2 is cut symmetrically through the center of a sphere of radius 2. Show that the volume removed is given by 1 F (x)dx, V= 0 where √ 1 F (x) = 4 3 − x2 + 4(4 − x2 ) arcsin √ . 4 − x2 √ √ Bonus: (*Difficult*) Show that V = 4 19π + 2 2 − 54 arctan 2 . 3 1 ...
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This note was uploaded on 01/11/2012 for the course MATH 100 taught by Professor Loveys during the Fall '11 term at McGill.

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