c Dr Oksana Shatalov, Spring 2013
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Spring 2013 Math 251
Week in Review 8
courtesy: Oksana Shatalov
(covering Sections 13.8,13.9,13.10 )
13.8: Triple Integrals
Key Points
• FUBINI’s THEOREM:If f is continuous on the rectangular box B = [a, b] × [c, d] × [r, s] then
ZZZ
Z sZ dZ b
f (x, y, z) dx dydz
f (x, y, z) dV =
r
B
c
a
and there are 5 other possible orders in which we can integrate.
• For solid region of TYPE I, E = {(x, y, z)|(x, y) ∈ D, φ1 (x, y) ≤ z ≤ φ2 (x, y)},
#
ZZZ
Z Z "Z
φ2 (x,y)
f (x, y, z) dV =
E
f (x, y, z) dz dA
D
φ1 (x,y)
• How to determine limits of integration for a solid region of type I1
1. Find an equation z = φ2 (x, y) for the upper surface and an equation z = φ1 (x, y) for the lower surface
of E. The functions φ1 (x, y) and φ2 (x, y) determine the lower and the upper z-limits of integration.
2. Make a 2-dimensional sketch of the projection D of E on the xy-plane to determine the limitsfor the
double integral over D.
Examples
1. Set up an iterated integral in rectangular coordinates for the volume of the solid bounded
by two elliptic paraboloids z = 3x2 + y 2 and z = 32 − 7x2 − y 2 .
2. Let E be the “wedge” in the first octant which is cut from the solid x2 + z 2 ≤ 4 by the
planes x = y and y = 0. Find mass of this solid if the density function is ρ(x, y, z) = z.
3. Find the volume of the solid E bounded by two parabolic cylinders z = x2 , z = 2 − x2 and
by two planes y = 0, y + z = 4.
4. Solid E is boundedRRRby the plane x = 9 and by the circular paraboloid x = y 2 + z 2 , set up
the triple integral E f (x, y, z) dV as an iterated integral in the order
(a) dx dz dy
(b) dz dy dx.
13.9: Cylindrical and Spherical Coordinates
Key Points
• The cylindrical coordinates
x = r cos θ
y = r sin θ
z=z
r ≥ 0, 0 ≤ θ ≤ 2π
are useful in problems that involve symmetry about the z-axis.
1
similar procedure works for the Type II and Type III regions.
c Dr Oksana Shatalov, Spring 2013
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• The spherical coordinates
x = ρ sin φ cos θ
y = ρ sin φ sin θ
z = ρ cos φ
ρ ≥ 0, 0 ≤ θ ≤ 2π,
0≤φ≤π
are especially useful in problems where there is symmetry about the origin.
5. Find the cylindrical coordinates of the point that has rectangular coordinates (4, −4, 7).
√
6. Find the spherical coordinates of the point that has rectangular coordinates (1, −1, 2).
7. Write the equation x2 + y 2 + z 2 = 10x in cylindrical and spherical coordinates
8. Write a description of the solid
p E in terms of inequalities involving spherical coordinates, if
E lies above the cone z = x2 + y 2 and inside the sphere x2 + y 2 + z 2 = 4z.
13.10: Triple Integral in Cylindrical and Spherical Coordinates
Key Points
• Let f (x, y, z) be a continuous function over a solid E ⊂ R3 . Let E ∗ be its image in cylindrical coordinates.
Then
ZZZ
ZZZ
f (x, y, z) dV =
f (r cos θ, r sin θ, z) dV ∗ ,
E∗
E
∗
where dV = r dr dz dθ.
• Let f (x, y, z) be a continuous function over a solid E ⊂ R3 . Let E ∗ be its image in spherical coordinates.
Then
ZZZ
ZZZ
f (x, y, z) dV =
f (ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ) dV ∗ ,
E∗
E
where dV ∗ = ρ2 sin φ dρ dθ dφ.
9. Let E be the region in the first octant bounded by the circular cone z 2 = x2 + y 2 and by
the plane z = 7. Set up the following triple integral in (a) cylindrical and (b) in spherical
coordinates:
ZZZ
f (x, y, z) dV
E
10. Set up the integral in (a) cylindrical and (b) spherical coordinates which gives the mass of
the ice-cream cone bounded above by the sphere x2 + y 2 √
+ z 2 = 16 and below by the cone
p
2
2
2
z = 3x2 + 3y 2 , if the density function is ρ(x, y, z) = x2 e x +y +z .
11. Find the center of mass of a solid right circular cone (the centroid of the cone) with height
H and base radius R assuming that the density is homogeneous.
12. Convert the integral
Z
1
Z √1−y2 Z √x2 +y2
xyz dz dx dy
0
x2 +y 2
0
to an integral in cylindrical coordinates, but don’t evaluate it.
Z Z √
Z √
9−y 2
3
9−x2 −y 2
(x2 + y 2 + z 2 ) dz dx dy to an integral in spheri-
13. Convert the integral
0
0
0
cal coordinates, but don’t evaluate it.
c Dr Oksana Shatalov, Spring 2013
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14. Convert the integral
Z
1
−1
Z √1−y2 Z
0
0
−
√
ln(4 + x2 + y 2 + z 2 ) dz dx dy
1−x2 −y 2
to an integral in spherical coordinates, but don’t evaluate it.