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(MATH100)quiz02-1a.pdf
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MATH 100 (L1) Introduction to Multivariable Calculus Spring 2006-07
. Quiz 02 (April 27, 2007): Multiple Integrals Duration: 40 minutes Name: ID No.: Tutorial Section: T1A
Answer ALL of the following 4 short answer questions. Show all your work for full credit.
1. (2 Marks) Sketch the solid G whose volume is given by the triple integral
Z 1 2 Z 2
1.x
Z
1 dy dz dx.
00 0
Answer:
2. (3 Marks) Evaluate the double iterated integral by interchanging the order of integration.
Z 1 Z 1
px3 +1 dx dy.
0 y
Answer: The iterated integral equals:
3. (2 Marks) Use polar coordinates to evaluate the double integral
ZZ 1
dx dy,
(1 + x2 + y2)2
D
where D = {x 2 + y 2 . a 2 : x . 0,y . 0}.
Answer: The double integral equals:
4. (3 Marks) Let G be the solid in the .rst octant bounded by the cylinders x 2 + y 2 = 1 and x 2 + y 2 = 6 and the planes z = 0, z = 1, x = 0, and x = y. Express the volume of the solid G as an iterated triple integral in cylindrical coordinates. You are NOT required to evaluate the triple integral but you have to write the integration limits properly.
Answer: The triple integral is: