We're trying to maximize (then minimize) the volume of a sphere wuth a cylinder drilled through the center. So I have a circle with the equation x^2 + y^2 = R^2, and I'm going to rotate it around the y-axis. Actually, I'm only going to rotate the quarter of it that lies in the first quadrant, then multiply its volume by 2. Now since the sphere has a cylindrical hole in it, my quarter-circle that I will rotate starts at sqrt(R^2-a^2), where a is half the height of the cylindrical hole. (sorry for the mess; I don't know how to draw a graph on the computer) SO. My integrand is (2)(pi)(x)(L(x))=(2)(pi)(x)sqrt(R^2-x^2) because the equation of a circle is y=sqrt(R^2-x^2) My upper and lower limits are R and sqrt(R^2-a^2) respectively. I didn't yet learn how to evaluate an integral of this form. Is there any way to find the max/min Volumes without evaluating the integal? Or is my integral wrong altogether?