Use this formula to derive the formula for triple integration in spherical coordinates SOLUTION Here the change of variables is given by We compute the Jacobian as follows: sin(Q) cos(8) | sin(q) sin(θ) cos(φ) -p sin(q) sin(θ) rho cos(φ) cos(8) rho cos(q) sin(θ) -p sin(φ) xu 0(p, θ, q) rho sin(p) cos(8) 0 - -p sin(4) sin(θ) rho sin(φ) cos(8) rho cos(φ) cos(8) rho cos(q) sin(8) -rho sin(φ)| sin(p) cos(θ) sin(φ) sin(8) -p sin(V) sin(θ) rho sin(φ) cos(8) = cos(p)(-22 sin(q) cos(p) sin2(8)-p? sin(4) cos(q) cos2(9) -rho sin(φ)(p sin-(φ) cos(θ) + rho sin-(φ) sin2(9) p2 sin(p) cos2(p) - --p2 sin(φ) Since 0 φ π, we have sin(φ)20. Therefore 0(p, θ, q) and this formula gives rx, y, z) dV- Kp sin(Q) cos(8), rho sin(Q) sin(θ), rho cos(q))p? sin(q) dp dθ dip which is equivalent to the formula for triple integration in spherical coordinates
