Homework 10 - APPM 2350 HOMEWORK 10(Full version Due Friday...

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APPM 2350 HOMEWORK 10 (Full version) FALL 2010 Due Friday, November 5, 2010 at 4 pm under your TA’s door. 1. Consider the finite solid object formed by ρ = sin(2 φ ), for 0 φ π / 2, with constant density δ . (a) Set up the integral(s) required to determine the z -coordinate of the center of mass using spherical coordinates in the order d ρ d φ d θ . (b) Repeat the calculation above, but using the order d φ d ρ d θ . (c) Evaluate one of your integrals. 2. When studying the formation of mountain ranges, geologists estimate the amount of work required to lift a mountain from sea level. Consider a mountain that is essentially in the shape of a right circular cone, with base radius R and height H . At some interior point located at ( x, y, z ), let the mass density of the material be δ ( x, y, z ), and let the height at the interior point be h ( x, y, z ). (a) Determine the definite triple integral that represents the total work done in forming the moun- tain. Your answer should be written in terms of R , H , δ ( x, y, z ), and h ( x, y, z ).
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