APPM 2350 Midterm: appm2350spring2016exam3_sol_1
Document Summary
This rst-octant region is described in cylindrical coordinates as. Z 2 cos r dz dr d = 4z /2. Perform u-substitution with u = 4 r2, du = 2r dr. r = 0 = u = 4 and r = 2 cos = u = 4 4 cos2 = Z 2 cos h(cid:0)4 sin2 (cid:1)3/2 sin3 d z /2. Z 4 sin2 rp4 r2 dr d = 2z /2. 0 u1/2 du d = 2z /2 (cid:0)sin3 1(cid:1) d (1 cos2 ) sin d . 2. (a) we are given that u = 3x + y and v = y + 4. Subtracting these yields u v = 3x 4 = x = 1. The second one itself gives y = v 4. (b) we need to substitute these variables into the original boundary. We"ll do it in pieces with brute force algebra: = u2 2uv + v2 + 8u 8v + 16. 2uv 8u 2v2 + 16v 32.
