Calculus
7th Edition
ISBN: 9781524916817
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 12.8, Problem 1PS
To determine
To discuss: the process of changing variables in a triple
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(10 points) Let f(x, y, z) = ze²²+y². Let
E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z ≤ 3}.
Calculate the integral
f(x, y, z) dv.
E
(12 points) Let
E={(x, y, z)|x²+ y² + z² ≤ 4, x, y, z > 0}.
(a) (4 points) Describe the region E using spherical coordinates, that is, find p, 0, and such
that
(x, y, z) (psin cos 0, psin sin 0, p cos) € E.
(b) (8 points) Calculate the integral
E
xyz dV using spherical coordinates.
(10 points) Let f(x, y, z) = ze²²+y². Let
E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z < 3}.
Calculate the integral
y,
f(x, y, z) dV.
Chapter 12 Solutions
Calculus
Ch. 12.1 - Prob. 1PSCh. 12.1 - Prob. 2PSCh. 12.1 - Prob. 3PSCh. 12.1 - Prob. 4PSCh. 12.1 - Prob. 5PSCh. 12.1 - Prob. 6PSCh. 12.1 - Prob. 7PSCh. 12.1 - Prob. 8PSCh. 12.1 - Prob. 9PSCh. 12.1 - Prob. 10PS
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- (14 points) Let f: R3 R and T: R3. →R³ be defined by f(x, y, z) = ln(x²+ y²+2²), T(p, 0,4)=(psin cos 0, psin sin, pcos). (a) (4 points) Write out the composition g(p, 0, 4) = (foT)(p,, ) explicitly. Then calculate the gradient Vg directly, i.e. without using the chain rule. (b) (4 points) Calculate the gradient Vf(x, y, z) where (x, y, z) = T(p, 0,4). (c) (6 points) Calculate the derivative matrix DT(p, 0, p). Then use the Chain Rule to calculate Vg(r,0,4).arrow_forward(10 points) Let S be the upper hemisphere of the unit sphere x² + y²+2² = 1. Let F(x, y, z) = (x, y, z). Calculate the surface integral J F F-dS. Sarrow_forward(8 points) Calculate the following line integrals. (a) (4 points) F Fds where F(x, y, z) = (x, y, xy) and c(t) = (cost, sint, t), tЄ [0,π] . (b) (4 points) F. Fds where F(x, y, z) = (√xy, e³, xz) where c(t) = (t², t², t), t = [0, 1] .arrow_forward
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