University:
Seattle Pacific University
Course:
MAT 1236 | Calculus III
Academic year:

2024
Views:

114

Pages:

1
Author:

Braiden Lam

Workshop #26 1. Sketch the region bounded by the paraboloids z = x2 + y 2 and z = 36 − 3x2 − 3y 2 . Use cylindrical coordinates to find the volume of this region. 2. Evaluate by first changing to cylindrical coordinates. (Sketch the region of integration! ) Z 4Z 0 √ 16−x2 Z 16−x2 −y 2 √ (x2 + y 2 ) dzdydx − 16−x2 0 3. Sketch the solid whose volume is given by the integral and evaluate. Z πZ πZ 0 ZZZ 4. Evaluate π 2 2 ρ2 sin φ dρdφdθ 1 (x2 + y 2 + z 2 )dV using spherical coordinates, where E is the region between the two E spheres ρ = 2 and ρ = 4 and above the cone φ = π in the first quadrant. 3 5. Using spherical coordinates, find the volume of the part of the sphere of radius 9 which is inside the π π sphere, between the cones φ = and φ = , and behind the yz-plane. 3 6