Calculus - Standalone book
3rd Edition
ISBN: 9781464125263
Author: Jon Rogawski, Colin Adams
Publisher: W.H. Freeman & Co
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Chapter 15.5, Problem 48E
To determine
Verify that every tangent plane to the cone
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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 15 Solutions
Calculus - Standalone book
Ch. 15.1 - Prob. 1PQCh. 15.1 - Prob. 2PQCh. 15.1 - Prob. 3PQCh. 15.1 - Prob. 4PQCh. 15.1 - Prob. 5PQCh. 15.1 - Prob. 1ECh. 15.1 - Prob. 2ECh. 15.1 - Prob. 3ECh. 15.1 - Prob. 4ECh. 15.1 - Prob. 5E
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Prob. 5ECh. 15.5 - Prob. 6ECh. 15.5 - Prob. 7ECh. 15.5 - Prob. 8ECh. 15.5 - Prob. 9ECh. 15.5 - Prob. 10ECh. 15.5 - Prob. 11ECh. 15.5 - Prob. 12ECh. 15.5 - Prob. 13ECh. 15.5 - Prob. 14ECh. 15.5 - Prob. 15ECh. 15.5 - Prob. 16ECh. 15.5 - Prob. 17ECh. 15.5 - Prob. 18ECh. 15.5 - Prob. 19ECh. 15.5 - Prob. 20ECh. 15.5 - Prob. 21ECh. 15.5 - Prob. 22ECh. 15.5 - Prob. 23ECh. 15.5 - Prob. 24ECh. 15.5 - Prob. 25ECh. 15.5 - Prob. 26ECh. 15.5 - Prob. 27ECh. 15.5 - Prob. 28ECh. 15.5 - Prob. 29ECh. 15.5 - Prob. 30ECh. 15.5 - Prob. 31ECh. 15.5 - Prob. 32ECh. 15.5 - Prob. 33ECh. 15.5 - Prob. 34ECh. 15.5 - Prob. 35ECh. 15.5 - Prob. 36ECh. 15.5 - Prob. 37ECh. 15.5 - Prob. 38ECh. 15.5 - Prob. 39ECh. 15.5 - Prob. 40ECh. 15.5 - Prob. 41ECh. 15.5 - Prob. 42ECh. 15.5 - Prob. 43ECh. 15.5 - Prob. 44ECh. 15.5 - Prob. 45ECh. 15.5 - Prob. 46ECh. 15.5 - Prob. 47ECh. 15.5 - Prob. 48ECh. 15.5 - Prob. 50ECh. 15.5 - Prob. 51ECh. 15.5 - Prob. 52ECh. 15.5 - Prob. 53ECh. 15.5 - Prob. 54ECh. 15.5 - Prob. 55ECh. 15.5 - Prob. 56ECh. 15.5 - 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Prob. 28ECh. 15.6 - Prob. 29ECh. 15.6 - Prob. 30ECh. 15.6 - Prob. 31ECh. 15.6 - Prob. 32ECh. 15.6 - Prob. 33ECh. 15.6 - Prob. 34ECh. 15.6 - Prob. 35ECh. 15.6 - Prob. 36ECh. 15.6 - Prob. 37ECh. 15.6 - Prob. 38ECh. 15.6 - Prob. 39ECh. 15.6 - Prob. 40ECh. 15.6 - Prob. 41ECh. 15.6 - Prob. 42ECh. 15.6 - Prob. 43ECh. 15.6 - Prob. 44ECh. 15.6 - Prob. 45ECh. 15.6 - Prob. 46ECh. 15.6 - Prob. 47ECh. 15.6 - Prob. 48ECh. 15.6 - Prob. 49ECh. 15.6 - Prob. 50ECh. 15.6 - Prob. 51ECh. 15.6 - Prob. 52ECh. 15.6 - Prob. 53ECh. 15.7 - Prob. 1PQCh. 15.7 - Prob. 2PQCh. 15.7 - Prob. 3PQCh. 15.7 - Prob. 1ECh. 15.7 - Prob. 2ECh. 15.7 - Prob. 3ECh. 15.7 - Prob. 4ECh. 15.7 - Prob. 5ECh. 15.7 - Prob. 6ECh. 15.7 - Prob. 7ECh. 15.7 - Prob. 8ECh. 15.7 - Prob. 9ECh. 15.7 - Prob. 10ECh. 15.7 - Prob. 11ECh. 15.7 - Prob. 12ECh. 15.7 - Prob. 13ECh. 15.7 - Prob. 14ECh. 15.7 - Prob. 15ECh. 15.7 - Prob. 16ECh. 15.7 - Prob. 17ECh. 15.7 - Prob. 18ECh. 15.7 - Prob. 19ECh. 15.7 - Prob. 20ECh. 15.7 - Prob. 21ECh. 15.7 - Prob. 22ECh. 15.7 - Prob. 23ECh. 15.7 - Prob. 24ECh. 15.7 - Prob. 25ECh. 15.7 - Prob. 26ECh. 15.7 - Prob. 27ECh. 15.7 - Prob. 28ECh. 15.7 - Prob. 29ECh. 15.7 - Prob. 30ECh. 15.7 - Prob. 31ECh. 15.7 - Prob. 32ECh. 15.7 - Prob. 33ECh. 15.7 - Prob. 34ECh. 15.7 - Prob. 35ECh. 15.7 - Prob. 36ECh. 15.7 - Prob. 37ECh. 15.7 - Prob. 38ECh. 15.7 - Prob. 39ECh. 15.7 - Prob. 40ECh. 15.7 - Prob. 41ECh. 15.7 - Prob. 42ECh. 15.7 - Prob. 43ECh. 15.7 - Prob. 44ECh. 15.7 - Prob. 45ECh. 15.7 - Prob. 46ECh. 15.7 - Prob. 47ECh. 15.7 - Prob. 48ECh. 15.7 - Prob. 49ECh. 15.7 - Prob. 50ECh. 15.7 - Prob. 51ECh. 15.7 - Prob. 52ECh. 15.7 - Prob. 53ECh. 15.7 - Prob. 54ECh. 15.7 - Prob. 55ECh. 15.7 - Prob. 56ECh. 15.7 - Prob. 57ECh. 15.7 - Prob. 58ECh. 15.7 - Prob. 59ECh. 15.7 - Prob. 60ECh. 15.8 - Prob. 1PQCh. 15.8 - Prob. 2PQCh. 15.8 - Prob. 3PQCh. 15.8 - Prob. 1ECh. 15.8 - Prob. 2ECh. 15.8 - Prob. 3ECh. 15.8 - Prob. 4ECh. 15.8 - Prob. 5ECh. 15.8 - Prob. 6ECh. 15.8 - Prob. 7ECh. 15.8 - Prob. 8ECh. 15.8 - Prob. 9ECh. 15.8 - Prob. 10ECh. 15.8 - 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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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