Problems: Spherical Coordinates - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.02SC | Multivariable Calculus
Academic year:

2023
Views:

401

Pages:

1
Author:

alinin4n

Problems: Spherical Coordinates 1.√ Find the volume of a solid spherical cap obtained by slicing a solid sphere of radius a 2 by a plane at a distance a from the center. (See picture.) Answer: In session 76 we found the limits: √ inner ρ: a/ cos φ to a 2, middle φ: 0 to π/4, outer θ: 0 to 2π. 2π π/4 √ a 2 ρ2 sin φ dρ dφ dθ. Volume = 0 1 Inner: ρ3 sin φ 3 √ a 2 a/ cos φ 0 a/ cos φ √ 2a3 2 a3 sin φ = sin φ − . 3 cos3 φ 3 √ √ √ π/4 2a3 2 a3 2a3 a3 2 2a3 a3 2 2a3 5a3 Middle: − cos φ − =− − − (− − )= − . 3 6 cos2 φ 3 3 3 6 3 6 φ=0 √ 2 2a3 5a3 a3 π √ Outer: 2π( − )= (4 2 − 5) ≈ 0.7a3 . 3 6 3 The volume of the entire sphere is about 12a3 and we’re looking at approximately the top sixth of its height. The sphere has more volume near its midpoint than at its top and bottom, so this answer seems reasonable.

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