calc 3 final exam - eastman
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7/27/2019 Calc 3 Final Exam - Eastman
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Math 5C -- Calculus III FINAL EXAMM. Eastman -- Spring 2008 page 2
8. Given , where consists of the line segment from to) a b a b a bBC .B B C .C ! !$ # a b# # # # ! #, followed by the arc of the circle from to , followed by the linesegment from to , verify Green's Theorem bya b a b! # ! !a. evaluating the line integral directly, and
b. evaluating the appropriate double integral.
9. Given ;F i j k B C D -9= D BC =38 C B Da b a b# #a bB / Da. Find curl . b. Find div .F F
Extra Credit
Given ; evaluateF i j k B C D -9= D BC =38 C B Da b a b# #a bB / Dwhere is the surface of the region bounded by the paraboloid ' ' f F SB C D . D Bf # #
and the plane , with normal vectors oriented outward. Justify your answer.D %
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Math 5C -- Calculus III FINAL EXAM -- ANSWERS
M. Eastman -- Spring 2008
1. Area # < .< . # % =38 % . # $' ' ' 1 11 1' '
# #
#
%=38#
)
) ) )" %# $a b 1
2. a. b. c.B $ &> C " %> D # > -9= B C D #"
" ## $
1
3. a. L > " # ! > # # " " #> # #> >
b. ' ' " "$ $# # # #> # .> #> .> ) 68$" "> >c. #*
4. a. b. c. d."#B C 'D $ %! "# $!BC "#C D $'B CD 'B $ $ # '(*5. Maximize subject to ; when0 B C D B C D "# 0 ' *"# B ' C % D #$ # 7+B
6. Use cylindrical coordinates: ' ' ' ! ! %# %) '%