21D-SSI08

# 21D-SSI08 - MAT21D Final Name: Student ID: Problem 1 2 3 4...

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MAT21D Final Name: Student ID: Problem Points Earned 1 8 2 7 3 6 4 12 5 10 6 10 7 10 8 10 Total 73

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2 Problem 1. Double integrals. (a) (2 points) Evaluate Z π 0 Z π x sin y y dy dx . (b) (3 points) A thin plate covers the region R bounded by y = x and y 2 = x in the ﬁrst quadrant. Assume the plate has constant density δ and mass M = 1. Find δ . (c) (3 points) Change to polar coordinates but do not evaluate. Z 2 0 Z 0 - 1 - ( y - 1) 2 xy 2 dx dy
Problem 2. Rectangular, Cylindrical and Spherical coordinates. (a) Set-up, but do not evaluate, volume integrals for the following solids. (i) (2 points) Cylindrical coordinates. D is the right circular cylinder whose base is the circle r = 3 cos θ in the xy -plane and whose top lies in the plane z = 5 - x . (ii) (2 points) Spherical coordinates. D lies above the cone φ = 3 π/ 4 and between the spheres x 2 + y 2 + z 2 = 9 and x 2 + y 2 + z 2 = 4. (b) (3 points)

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## This note was uploaded on 05/19/2009 for the course MATH 21d taught by Professor Dianwenzhu during the Spring '07 term at UC Davis.

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21D-SSI08 - MAT21D Final Name: Student ID: Problem 1 2 3 4...

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