12-08-cylindrical-spherical-integrals

12-08-cylindrical-spherical-integrals - Math 21a...

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Unformatted text preview: Math 21a Cylindrical & Spherical Coordinates Spring, 2009 When we change from Cartesian coordinates ( x,y,z ) to cylindrical coordinates ( r,θ,z ) or spherical coordinates ( ρ,θ,φ ), integrals transform according to the rule dV = dxdy dz = r dr dθ dz = ρ 2 sin φdρdθ dφ. 1 Using cylindrical coordinates, evaluate the integral RRR E p x 2 + y 2 dV , where E is the solid in the first octant inside the cylinder x 2 + y 2 = 16 and below the plane z = 3. 2 Sketch the solid whose volume is given by the integral R π/ 2 R 2 R 9- r 2 r dz dr dθ, and evaluate the integral. 3 Use spherical coordinates to evaluate RRR E z dV , where E lies between the spheres x 2 + y 2 + z 2 = 1 and x 2 + y 2 + z 2 = 4 in the quarter-space where y ≤ 0 and z ≥ 0. 4 Use spherical coordinates to set up a triple integral expressing the volume of the “ice-cream cone,” which is the solid lying above the cone φ = π/ 4 and below the sphere ρ = cos φ . Evaluate it. 5 Sketch the region of integration for Z 1 Z √ 1- x 2 Z √ 2- x 2- y 2 √ x 2 + y 2 xy dz dy dx, and evaluate the integral by changing to spherical coordinates....
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12-08-cylindrical-spherical-integrals - Math 21a...

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