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You have ﬁfteen minutes to complete the quiz. If anything seems unclear,
please ask.
1. Let f (x, y, z ) be a continuous function of three variables. Let R be the
region between the planes x = 0, y = x, z = 0 and z = 1 − y . Write the
integral of f over R in the orders: (a) dzdydx; (b) dydxdz . (2 points each) 2. Let R be the region where z ≥ 0, inside the sphere center the origin and
radius 2 and outside the sphere, center the origin and radius 1. Set up
triple integrals for the volume of f in: (a) cylindrical coordinates; (b)
spherical coordinates. (2 points each)
Use one of your solutions to (a) and (b) to compute the volume of R. (2
points) 1 Solutions
If you have any questions and / or think you’ve spotted a mistake / typo, please
get in touch with me.
1. (a)
1−y 1 1 f (x, y, z )dzdydx.
0 0 x 1 1−z (b)
1−z f (x, y, z )dydxdz.
2. (a)
2π √ 2 x 0 0 4−r 2 2π rdzdrdθ +
0 1 0 0 √ 1
0 √ 4−r 2 rdzdrdθ.
1−r 2 (b)
2π π /2 2 ρ2 sin(φ)dρdφdθ.
0 0 1 The second of these is the easier to compute (although the ﬁrst is possible).
The inner integral computes to
7 sin(φ)
,
3
the middle to 7π
,
3
and the outer to 14π/3, which is thus the volume. 2 ...
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 Spring '08
 ALDROUBI
 inner integral computes

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