MA 242 - Fall 2010
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Worksheet VI
Triple Integrals
Rectangular Coordinates
ZZZ
1. (In Class) Evaluate
xy dV where E is the domain of integration/region in R3
E
that lies under the graph of f (x, y) = x2 + y 2 + 4 and over the rectangular region
[0, 2] × [0, 1] in the xy-plane.
Figure 1: The domain of integration for Problem 1.1 (Left) and Problem 1.2 (Right).
ZZZ
2. (In Class) Evaluate
xy dV where E is the domain of integration/region in R3
E
that lies under the graph of f (x, y) = x2 + y 2 + 4 and over the region in the xy-plane
bounded by the curve y = x2 and the lines y = 0, x = 0 and x = 1.
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ZZZ
3. Evaluate
xy dV where E is the domain of integration/region in R3 that lies under
E
the graph of√f (x, y) = x2 + y 2 + 4 and over the region in the xy-plane bounded by the
curves y = x, y = 0 and x = 1.
Figure 2: The domain of integration for Problem 1.3.
ZZZ
4. Evaluate
ex y dV where E is the domain of integration bounded by the parabolic
E
cylinder z = 1 − y 2 and the planes z = 0, x = 0 and x = 2. (Hint: To determine the
bounds for the region with respect to y, determine where z = 1 − y 2 is positive).
Figure 3: The domain of integration for Problem 1.4.
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ZZ
5. (In Class) Evaluate
xey dV where E is the domain of integration bounded by the
E
parabolic cylinder z = 1 − x2 and the the planes z = 0, y = 1 and y = −1.
Figure 4: The domain of integration for Problem 5.
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Interlude for Strategy on Changing Coordinates
ZZ
Below is a suggested strategy for changing coordinate systems and completing double(
f (x, y) dA
D
ZZZ
f (x, y, z) dV ) integrals.
or triple(
D
1. Decide an appropriate coordinate system. Typically, this is based off of the shape of
the domain of integration.
2. Convert the integrand f (x, y) or f (x, y, z) to the chosen coordinate system representation of the function.
3. Express dA or dV in the chosen coordinate form.
4. Describe the domain of integration in the chosen coordinate system.
5. Decide an appropriate order of integration and complete the problem.
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Cylindrical Coordinates
ZZZ
z
1. (In Class) Evaluate
p
x2 + y 2 dV where E is the region/domain of integration
E
in R3 bounded by the cylinder x2 + y 2 = 4 and the planes z = 0 and z = 5.
Figure 5: The domain of integration for Problem 3.1
ZZZ
2. (In Class) Evaluate
ez dV where E is the region/domain of integration in R3
E
bounded by the cylinder x2 + y 2 = 5, the paraboloid z = 1 + x2 + y 2 and the plane
z = 0.
Figure 6: The domain of integration for Problem 3.2
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ZZZ
3. Evaluate
x3 + xy 2 dV where E is the region/domain of integration in R3 that
E
lies in the first octant (x ≥ 0, y ≥ 0, z ≥ 0) and beneath the paraboloid z = 3 − x2 − y 2 .
(Hint: Factor the function f (x, y, z) = x3 + xy 2 before converting the function to its
cylindrical coordinate representative.)
Figure 7: The domain of integration for Problem 3.3
4. Find the volume of the solid that lies within both the cylinder x2 + y 2 = 1 and the
sphere x2 +y 2 +z 2 = 5. (Hint: This problem can be handled in two different ways, both
which agree. i.) You can use a doublepintegral and polar coordinates and integrate an
appropriate function (z = f (x, y) = 5 − x2 − y 2 ) over an appropriate region in the
xy-plane, or ii.) You can use a triple integral and cylindrical coordinates and integrate
the function f (x, y, z) = 1 over the appropriate region in R3 .)
Figure 8: The domain of integration for Problem 3.4 (Left) and the two surfaces (right).
The sphere provides caps for the cylinder.
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Spherical Coordinates
Remark 4.1. When comparing class notes with your textbook, the two symbols φ and ϕ are
equal. They both represent the angle between the radial line segment and the positive z-axis.
φ = ϕ.
ZZZ
(4 − x2 − y 2 − z 2 ) dV , where the domain of integration E is the solid
1. Evaluate
E
hemisphere x2 + y 2 + z 2 ≤ 9, z ≥ 0. (Hint: Recall in our spherical coordinates that
the angle ϕ determines whether we are above/on/below the xy-plane.)
Figure 9: The domain of integration for Problem 4.1
ZZZ
2. Evaluate
z dV, where the domain of integration E is the region ...
E
(a) that is bounded by the sphere x2 + y 2 + z 2 = 1.
(b) that lies in the first octant (x ≥ 0, y ≥ 0, z ≥ 0) and inside the sphere x2 +y 2 +z 2 =
1.
(c) that lies in the first octant (x ≥ 0, y ≥ 0, z ≥ 0) and between the spheres x2 +
y 2 + z 2 = 1 and x2 + y 2 + z 2 = 4.
Remark 4.2. This is essentially an exercise in describing regions in spherical
coordinates and setting up triple integrals in spherical coordinates. The integration
remains the same from part to part.
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Figure 10: The domains of integration for Problem 4.2.a (Left) , 4.2.b (Middle), and 4.2.c
(Right)
ZZZ
3. Evaluate
e(x
2 +y 2 +z 2
3
) 2 dV where the domain of integration E is the solid quarter-
E
sphere x2 + y 2 + z 2 ≤ 9 and y ≥ 0, z ≥ 0.
Figure 11: The domain of integration for Problem 4.3
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