"A round hole of radius is bored through the center of a sphere of radius 2 feet. Find the volume of the piece cut out."
I have a vague idea on how to do this. . Integral of the surface area. Except, I don't know what to do with the extra variable.
Draw a 2-dimensional picture of a cross-section through the middle. Look at the right hand side. You have a 2-dimensional object bounded on the right by the curve and on the left by the line . The points where these curves intersect are and . We want the volume of the solid of revolution formed by rotating this about the y axis.
Method 1 (the shell method):
Slice the 2-dimensional object into vertical strips. Each such vertical strip is located at position x, has width dx, and height . Rotating this strip about the vertical axis gives a thin cylindrical shell. Slice this open and lay it out flat. The result is a rectangular slap of width (the circumference of the shell), height , and thickness dx so and
Method 2 (the "washer" method):
Slice the 2-dimensional object into horizontal strips. Each such horizontal strip is located at position y and has width dy. Its left endpoint is at and its right endpoint is at . Rotating this strip about the vertical axis gives a thin flat "washer" of thickness dy. The inner radius of this (annulus) is and the outer radius is . The volume is
I'm sure you know how to solve either of these integrals.