Consider the vector D = 3 sin ⁡ ( θ ) ar (unit vector).Evaluate the integral ∬ S D d S ,where S is the surface of a sphere with radius r=5 centered at the origin.

Question

Consider the vector   D  =  3  sin  ⁡  (  θ  ) ar (unit vector).Evaluate the integral      ∬    S    D     d  S  ,where S is the surface of a sphere with radius r=5 centered at the origin.

Donovan Shields · Accepted Answer

In terms of polar coordinates, the surface area element at constant r, in your case r=5, is       r    2    sin  ⁡  θ  d  θ  d  ϕ. Let             n      ^       denote the unit vector parallel to, D so your integral is      ∫    0          2      π        d  ϕ      ∫    0    π    d  θ      r    2    sin  ⁡  θ  D  ⋅            n      ^        =  150  π      ∫    0    π        sin    2    ⁡  θ  d  θ  =  75      π    2    .

this is my question, any help would be appreciated. Consider the vector D=3sin( theta) ar (unit vector).

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