(e) The right hand side becomes -mgart (p) . (upAt) = -mgart (p) . (-Ap) , where -Ap is the change in height in At and the additional negative sign is from the opposite dependence between height and radius, namely the higher the ball is, the shorter the the distance from the center is. Recalling the very definition, this quantity is the work done by artificial gravitational force within At. If we sum all the contributions up from p = R to p = r, the total work done by the artificial gravitational force can be written as the following integral: W = + ma(p)ap = - "mg ( p ) dap