Use multiple integrals to find the volume of the solid E that lies above the con
ID: 1947651 • Letter: U
Question
Use multiple integrals to find the volume of the solid E that liesabove the cone z= sqt (x^2 + y^2 ) and below the sphere x^2 + y^2 + z^2 =9
Explanation / Answer
Using spherical coordinates: z = v(x^2/3 + y^2/3) ==> ? cos f = v(1/3) ? sin f ==> tan f = v3 ==> f = p/3. So, the volume ???E 1 dV equals ?(? = 0 to 2p) ?(f = 0 to p/3) ?(? = 0 to 4) 1 * (?^2 sin f d? df d?) = 2p ?(f = 0 to p/3) sin f df * ?(? = 0 to 4) ?^2 d? = 2p [-cos f {for f = 0 to p/3}] * [(1/3) ?^3 {for ? = 0 to 4}] = 2p (1 - 1/2) * (64/3) = 64p/3. Note: The centroid integrals are done similarly (same bounds, and convert the cartesian vestiges to spherical coordinates.)
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