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Test 3 Review
1. Evaluate the following integrals (you may need to switch order of integration):
(a)
Z
π
0
Z
x
0
x
sin
y dy dx
(b)
Z
3
0
Z
1
√
x/
3
e
y
3
dy dx
(c)
Z
1

1
Z
√
1

y
2

√
1

y
2
(
x
2
+
y
2
)
dy dx
(d)
Z
2
0
Z
√
1

(
x

1)
2
0
x
+
y
x
2
+
y
2
dy dx
(e)
Z
1
0
Z
2

x
0
Z
2

x

y
0
dz dy dx
(f)
Z
1

1
Z
√
1

y
2
0
Z
x
0
(
x
2
+
y
2
)
dz dx dy
2. Set up integrals to integrate over the following regions:
(a) the solid whose base is the region in the
xy
plane that is bounded by the parabola
y
= 4

x
2
and
the line
y
= 3
x
, while the top of the solid is bounded by the plane
z
=
x
+ 4.
(b) the (2dimensional) region bounded by
y
=
e
x
and the lines
y
= 0
,x
= 0
,x
= ln 2.
(c) the (2dimensional) region bounded by the parabolas
x
=
y
2

1 and
x
= 2
y
2

2.
(d) the region cut from the ﬁrst quadrant by the cardiod
r
= 1 + sin
θ
.
(e) the tetrahedron cut from the ﬁrst octant by the plane 6
x
+ 3
y
+ 2
z
= 6.
(f) the region bounded by the paraboloids
z
= 8

x
2

y
2
and
z
=
x
2
+
y
2
.
(g) the region bounded by the cylinder
z
=
y
2
, and the planes
z
= 0
,x
= 0
,x
= 1
,y
=

1
,y
= 1.
(h) (using cyclindrical coordinates) region bounded by
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This note was uploaded on 05/24/2011 for the course MTH 234 taught by Professor Irinakadyrova during the Spring '10 term at Michigan State University.
 Spring '10
 IrinaKadyrova
 Integrals

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