MAT368H5S
Term Test Solutions
1. [5 marks] Use polar coordinates to evaluate
2
(ln 2) y 2
ln 2
exp ( x2 + y 2 )dxdy.
0
0
Solution. We rst sketch the region D of integration determined by the limits of the iterated integrals.
(ln 2)2 y 2
ln 2
/2
x2
exp (
MAT368H5S
Quiz 3
Tuesday, March 5, 2013
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Student #:
Question 1 [5 marks] Consider the volume of the region bounded above by the sphere x2 + y 2 + z 2 = 64
and below by the cone z = x2 + y 2 .
(a) Set up the integral usi
MAT368H5S
Quiz 2
Feb 5, 2013
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Question 1 [5 marks] Evaluate D ydA, where D is the region in the rst quadrant that lies between the
circles x2 + y 2 = 4 and x2 + y 2 = 2x.
Solution.
x2 + y 2 = 4 r = 2
x2 + y 2
MAT368H5S
Quiz 1
Jan 22, 2013
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Tutorial:
Question 1 [5 marks] Evaluate
Solution.
xy
xy
R
x2 + y 2 + x sin(x + 2y ) dA, where R = [0, 1] [0, 1].
x2 + y 2 + x sin(x + 2y ) dA =
xy
x2 + y 2 dA +
R
R
1
x2
xy
+
y
MAT368H5S-2013
Homework 2
1. [5 marks] Let E be the region in the rst octant of 3-space of nite volume that is bounded by the
surfaces z = 1, y = 0, x + y = 1, and z = y 2 . Express the integral of f (x, y, z ) over E in the six different
orders of integr