Evaluate 0pi 0pi2 0r x2sintheta dx dtheta dphiSolutionEvalua

Evaluate ₀^pi ₀^pi/2 ₀^r x²*sin(theta) dx d(theta) d(phi)

= ₀^pi ₀^pi/2sin(theta)[ ₀^r x² dx] d(theta) d(phi)

= ₀^pi ₀^pi/2sin(theta)[x³/3 ]₀^r d(theta) d(phi)

= r³/3 ₀^pi [₀^pi/2sin(theta) d(theta)] d(phi)

= r³/3 ₀^pi [-cos(theta)]₀^pi/2 d(phi)

= -r³/3 ₀^pi [cos(pi/2)-cos(0)] d(phi)

= -r³/3 ₀^pi [-1)] d(phi)

= r³/3 ₀^pi d(phi)

= r³/3 [phi)]₀^pi

= r³/3 [pi - 0)]

= pi*r³/3

thus

₀^pi ₀^pi/2 ₀^r x²*sin(theta) dx d(theta) d(phi) = pi*r³/3