Quiz8-4_5

# Quiz8-4_5 - Name You have ﬁfteen minutes to complete the...

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Unformatted text preview: Name: 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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Quiz8-4_5 - Name You have ﬁfteen minutes to complete the...

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