Let S_1 denote the surface defined by the equation x^2 + y^2 + z^2 = 4z and S_2

Question

Let S_1 denote the surface defined by the equation x^2 + y^2 + z^2 = 4z and S_2 denote the surface defined by the equation x^2 + y^2 = z. (a) Give specifics on the geometries of S_1 and S_2. (b) Give the equations of S_1 and S_2 in spherical coordinates. (c) Set up, but not evaluate, a triple integral in spherical coordinates. which represents the total electrical charge of the solid region between S_1 and S_2. which is electrically charged with the charge density given by the equation sigma(x, y, z) = z e^xy.

Explanation / Answer

s1 = sphere and s2= paraboloid

In spherical co-orinates

S1 = r^2 = 4*r*cos(theta)

r = 4*cos(theta)

S2 = r^2 sin^2 (theta) =rcos(theta)

r*sin(theta)*tan(theta) = 1

(c)Triple integral will be integral (del.sigma) dV ------------- over total volume of both surfaces

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