Section 15.6 (6th Ed): Exercise 6:

assume(z > 0);

integrate(integrate(integrate(z*exp(-y^2),x,0,y),y,0,z),z,0,1);

Exercise 10: Integrate over E = {(x,y,z): 0<x<1, 0<y<x, x<z<2x}

integrate(integrate(integrate(y*z*cos(x^5),y,0,x),z,x,2*x),x,0,1);