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Math 237
Assignment 9
Not to be handed in
1.
Use
T
(
x,y
) = (
x
+
y,

x
+
y
) to evaluate
R
π
0
R
π

y
0
(
x
+
y
) cos(
x

y
)
dx dy
.
2.
Find a linear transformation that maps
x
2
+ 4
xy
+ 5
y
2
= 4 onto a unit circle. Hence
show that the area enclosed by the ellipse equals 4
π
.
3.
Evaluate
RRR
R
x
2
+
y dV
where
R
is the region bounded by
x
+
y
+
z
= 2,
z
= 2,
x
= 1
and
y
=
x
.
4.
Evaluate
RRR
R
xyz
e
x

y
+
z
dV
, where
R
xyz
is bounded by the planes
x

y
+
z
= 2,
x

y
+
z
= 3,
x
+ 2
y
=

1,
x
+ 2
y
= 1,
x

z
= 0 and
x

z
= 2.
5.
Show that the region
D
in the ﬁrst quadrant bounded by
ay
=
x
3
,
by
=
x
3
,
cx
=
y
3
and
dx
=
y
3
has area
1
2
(
√
b

√
a
)(
√
d

√
c
).
6.
Evaluate the following integrals.
a)
RRR
D
(
x
2
+
y
2
+
z
2
)
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This note was uploaded on 01/04/2012 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.
 Spring '08
 WOLCZUK
 Unit Circle

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