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(MATH100)quiz02-1c.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: T1C
Answer ALL of the following 4 short answer questions. Show all your work for full credit.
1. (2 Marks) Suppose that the area of a region in polar coordinate system is given by the double integral
Z 3/4 Z 2 sin
r dr d,
/4 csc
where csc =1/ sin . Sketch and shade the region and .nd its area.
Answer: The area of the region is:
2. (3 Marks) Evaluate the double iterated integral by interchanging the order of integration.
Z 2 Z 1 x 3
ye dx dy.
0 y/2
Answer: The iterated integral equals:
3. (2 Marks) The rectangle R in the xy-plane consists of those points (x, y) for which 0 . x . 2 and 0 . y . 1. Find the volume of the solid that lies below the surface z =1+ xy and above R.
Answer: The volume of the solid is:
4. (3 Marks) Let G be the solid that is bounded below by the cone z = p(x2 + y2)/3 and bounded
222
above by the sphere x + y + z = z. Express the volume of the solid G as an iterated triple integral in spherical coordinates. You are NOT required to evaluate the triple integral but you have to write the integration limits properly.
Answer: The triple integral is: