1. Collect homework.
2. Recall that to find the area of a plane region, we divide the region into thin rectangles, add the areas of rectangles to form a Riemann sum, and then take the limit of the Riemann sums to obtain an integral for the area:
3. This week, we are going to examine volumes of solids of revolution. We use the same strategy to find the volume of a solid. We will divide the solid into thin slabs, approximate the volume of each slab, add the approximations together to form a Riemann sum, and then take the limit of the Riemann sums to form an integral for the volume of the solid.
In our first method, we will be summing up volumes of disks (or cylinders).
Show Styrofoam circles on a dowel to illustrate this idea.
4. Example
5. One of the most difficult things to do when working with volumes of solids of revolution is to visualize the shape that is being formed.
To help with this visualization process, use one of the following techniques:
(A) Use styrofoam disks
Click here for a picture
(B) Use power drill with Styrofoam cut-outs mounted on dowels. When the styrofoam rotates it traces out the solid of revolution.
Click here for a picture
(C) Do some edible calculus (by Nancy Dirnberger): (USE after SHELL METHOD) If you core an apple you have a great solid of revolution with a hole through the solid! An apple is usually sliced in one of two ways. If you slice it so that the plane of the slice contains what would be the axis of revolution, your apple slice (actually you have two) is a disc of revolution. You can almost slice it thin enough to have width dx! If you slice the apple into rings, the resulting slices are washers that will give you the same volume when added. With a large, thick slice of a Bermuda onion you can pull up onion rings as successive cylindrical shells. Students often have a problem visualizing these cylindrical shells.
6. Determine the volume of a sphere.
7. Example (with respect to the y-axis)
