The question is as follows:

a: Find the center of mass of a solid of constant density bounded below by the paraboloid $\displaystyle z^2 =x^2 + y^2 $

and above by the plane z=4

b. Find the plane, z=c, that divides the solid into two parts of equal

volume. This plane does not pass through the center of mass.

I already found part A to be (0,0,8/3). My question is how come the plane z=8/3 doesn't split the figure into 2 equal volumes? It's the center of mass point in the z direction AND the density is uniform. Because D=M/V $\displaystyle \Rightarrow $ and D is constant, [tex] V=cM [tex].

To make matters more confusing (to me), my math textbook (in the solution) simply sets M/2 (which is 4pi) equal to the integral with upper limit limit of z as C. Isnt the center of mass supposed to be the point where it in the center of the weighted distribution of the mass? I attached a picture of the textbooks solution to better illustrate my point. The problem I am confused about is 25 b).

Thank You.