高微問題(change of variable)

find the mass of a ball of radius R if the mass density is c times the distance

from the boundary of the ball

1 個解答

• 1 0 年前
最佳解答

Without loss of generality, the ball is centered at the origin : B={(x,y,z)|x^2+y^2+z^2=R^2} = {(rho, theta, phi)|0<rho<R, 0<theta<2pi, 0<phi<pi}.

the distance from the boundary of the ball =R-rho, if the latter (sherical coordinates) is used.

Thus the mass of a ball , M=c* triple integral[over B] of {R-rho} dV. To finish the calculation, recognize dV=(rho)^2*sin(phi)*d(rho)d(theta)d(phi) and change the triple integral into iterated integral with the order d(rho) the last will do.