The simplest way to do this is to first find the volume of the entire sphere, , then subtract the volume removed. That will be a cylinder plus two "caps" at either end. The radius of the cylinder is, of course, 2 cm. To find the height, h, draw a circle with a thin rectangle inscribed in it. The line from the center of the circle to a corner of the rectangle has length the radius of the sphere, 4 cm. A line from the center of the sphere perpendicular to the side of the rectangle has length the radius of the cylinder, 2 cm. Use the Pythagorean theorem to find half the length of the cylinder. Two find the volume of a cap, think of it as a stack of disks. Setting up a coordinate system with the origin at the center of the sphere, the equation of the sphere's projection in the xy-plane is or [itex]x= \pm\sqrt{16- y^2}[/tex] and that is the radius of one of the disks. Its area is . If its thickness is dx, the volume of one disk is and so the volume of them all is [tex]\pi \int 16- y^2 dy[tex]. The integral is take from the top of the cylinder to the top of the sphere.

I would do this in two parts.2) Consider that part of the parabola y = (x-1)*(x+2) which lies below the x-axis. Calculate the volume of the solid formed when this part is rotated about the line y =4.

First look at the parabola only. Imagine a line from to y= 4. Rotating around the axis y= 4 gives a disk of radius and so of area . Taking each disk to have "thickness" dx, The volume of each disk is and the volume of all put together is . The integral is, of course, from x= -2 to x= 1 where the parabola crosses the x-axis.

Second, subtract the volume of the cylinder made up of the portion that is above the x-axis. That will be a cylinder with radius 4 and height 1-(-2)= 3.