APPM 2350 Midterm: appm2350summer2016exam3_sol
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E2x sin(cid:0) y2(cid:1) sin(cid:0) y2(cid:1) dx dy dz =z 1 dz dy = 4 z 1 y2. 0 z 1 y sin(cid:0) y2(cid:1) dy ln 3. 2 sin u du = 4 sin(cid:0) y2(cid:1) y2 dy dz (cid:4: (20 pts) consider the solid between two concentric spheres of radii r1 and r2 with 0 < r1 < r2. The mass density of the solid varies inversely with the cube of the distance from the center of the spheres, that is, mass density = k/(distance3). Use spherical coordinates, placing the origin at the center of the concentric spheres. Then the density can be written as k. Total mass = m =zzz dv =z 2 . 3 2 sin d d d k d (cid:19) = 2 cos (cid:12)(cid:12)(cid:12) Zzzq z dv, p1 + x2 + y2. The integrand becomes z p1 + x2 + y2 z.
