Old_Final

# Old_Final - then I γ F · d r = 0 for any closed loop γ ....

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32B Killip Practice Final First Name: Last Name: Section: There are THIRTEEN problems; ﬁve points per problem. Rules. No calculators, computers, notes, books, crib-sheets,. .. Out of consideration for your class-mates, no chewing, humming, pen- twirling, snoring,. .. Try to sit still. The answers do not involve nasty integrals; if you arrive at one, check your work. 1 2 3 4 5 6 7 8 9 10 11 12 13 Σ

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(1) Determine the surface area of the paraboloid x 2 + y 2 = 2 z, 0 z 1 by whatever means you wish.
(2) Calculate the integral Z R x 2 dA where R is the region below: - 6 2 1 2 1 R ± ± ± ± ± ± ± ± ± ± ± ² ² ² ² ² ² ² ² ² ² ²

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(3) State the Fundamental Theorem for Line integrals
(4)+(5) Compute both sides of the Divergence Theorem for the cylinder x 2 + y 2 1 0 z 1 with F = z sin( x 2 + y 2 ) k . (Of course, they should turn out to be equal).

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[Extra space for (4)+(5)]
(6) Determine Z R z dV where R is the intersection of the cone x 2 + y 2 z 2 and the unit ball x 2 + y 2 + z 2 1.

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(7) Justify the following statement: If ∇ × F = 0

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Unformatted text preview: then I γ F · d r = 0 for any closed loop γ . (8) Consider the region R given by ≤ z ≤ ( y-x 2 ) 2 , x 2 ≤ y ≤ x. Use the change of variables x = u, y = v + u 2 , z = wv 2 , to evaluate Z R dV y-x 2 (9) Let R be the region where 0 ≤ y ≤ x and x 2 + y 2 ≥ 1. Evaluate Z R dA ( x 2 + y 2 ) 2 by switching to polar coordinates. (10) Which of the following is conservative: F = x i + e y j + xe y k or F = ye x i + e x j + z k Write it as ∇ f . (11) Evaluate Z C cos( πx ) dx + xdy where C is the following loop traversed clockwise-6 2 1 2 1 (12) Determine Z dS √ x 2 + z 2 over the oblique cone parameterized by x = v + v cos( u ) , y = v sin( u ) , z = v for u ∈ [0 , 2 π ] and v ∈ [0 , 1]. (13) Compute the following integral by reversing the order Z 1 Z y y 2 1 1-√ x dxdy...
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## This note was uploaded on 06/03/2011 for the course PHYSICS 1B 1b taught by Professor Corbin during the Winter '10 term at UCLA.

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Old_Final - then I γ F · d r = 0 for any closed loop γ ....

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