In Section 8.5 we calculated the center of mass by considering objects composed of ajinite number of point masses or objects that, by symmetry, could be represented by a finite number of point masses. For a solid object whose mass distribution does not allow for a simple determination of the center of mass by symmetry, the sums of Eqs (8.28) must be generalized to integrals where x and y are the coordinates of the small piece of the object that has mass t/m the integration is over the whole of the object. Consider a thin rod of length L, mass M, and cross-sectional area A. Let the origin of the coordinates be at the left end of the rod and the positive x-axis lie along the rod.
(a) If the density p = M/V of the object is uniform, perform the integration described above to show that the x-coordinate of the center of mass of the rod is at its geometrical center.
(b) If the density of the object varies linearly with x--that is, p = ax, where a is a positive constant€”calculate the x-coordinate of the rod's center of mass.

x dm dm