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# exam1 - R 1 R 1 √ y R 1-x f x y z dzdxdy in dxdzdy order...

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Calculus IV S1202Q Section 002, Summer 2007 Exam 1 Thomas D. Peters Name: July 19, 2007 Do all problems, in any order. Show your work. An answer alone may not receive full credit. No notes, texts, or calculators may be used on this exam. Problem Possible Points Points Earned 1 15 2 12 3 12 4 12 5 12 6 12 7 10 TOTAL 85 1

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1. (15 points) a) Compute 1 0 1+ y - y y dxdy. b) Write this iterated integral as a sum of one or more integrals in dydx order. You are not asked to evaluate it this way! Answer: a) 1 / 2 + 2 / 5 + 1 / 3 b) 0 - 1 1 - x y dydx + 1 0 1 0 y dydx + 2 1 1 ( x - 1) 2 y dydx 2. (12 points) Find the volume of the solid which is between the paraboloids z = x 2 + y 2 - 1 and z = 5 - 2 x 2 - 2 y 2 . Answer: 6 π 3. (12 points) Evaluate the integral 3 0 9 y 2 y cos( x 2 ) dxdy . Answer: sin 81 4 4. (12 points) Evaluate the integral 3 - 3 9 - x 2 9 - x 2 9 - x 2 - y 2 0 z x 2 + y 2 + z 2 dzdydx Answer: 3
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Unformatted text preview: R 1 R 1 √ y R 1-x f ( x, y, z ) dzdxdy in dxdzdy order. Answer: R 1 R 1-√ y R 1-z √ y f ( x, y, z ) dxdzdy 6.(12 points) Calculate the volume of the ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 using a suitable change of coordinates on R 3 . Answer: 4 3 abc 7. (10 points) Find a region E for which the triple integral RRR E (1-x 2-2 y 2-3 z 2 ) dV is maximum. Brieﬂy explain. Answer: E = { ( x, y, z ); 1-x 2-2 y 2-3 z 2 ≥ } , ie the interior of the ellipsoid x 2 + 2 y 2 + 3 z 2 = 1. The integral is maximized there since this is the only place where the integrand, 1-x 2-2 y 2-3 z 2 , is non-negative. 2...
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