Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena College Photo by Vickie Kelly, Day 3 The Shell Method and Arc length Japanese Spider Crab Georgia Aquarium, Atlanta

Find the volume of the region bounded by,, and revolved about the y - axis. We can use the washer method if we split it into two parts: outer radius inner radius thickness of slice cylinder Japanese Spider Crab Georgia Aquarium, Atlanta

If we take a vertical sliceand revolve it about the y-axis we get a cylinder. cross section If we add all of the cylinders together, we can reconstruct the original object. Here is another way we could approach this problem:

cross section The volume of a thin, hollow cylinder is given by: r is the x value of the function. h is the y value of the function. thickness is dx.

cross section If we add all the cylinders from the smallest to the largest: This is called the shell method because we use cylindrical shells.

Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal slice directly.

Shell method: If we take a vertical slice and revolve it about the y-axis we get a cylinder.

Note:When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.

When the strip is parallel to the axis of rotation, use the shell method. When the strip is perpendicular to the axis of rotation, use the washer method. 

If we want to approximate the length of a curve, over a short distance we could measure a straight line. By the pythagorean theorem: We need to get dx out from under the radical. Length of Curve (Cartesian)Lengths of Curves:

Example: Now what? This doesn’t fit any formula, and we started with a pretty simple example! The TI-89 gets:

Example: The curve should be a little longer than the straight line, so our answer seems reasonable. If we check the length of a straight line:

Example: You may want to let the calculator find the derivative too: Important: You must delete the variable y when you are done! ENTER F44 Y STO Y

Example:

If you have an equation that is easier to solve for x than for y, the length of the curve can be found the same way. Notice that x and y are reversed. ENTER X STO

Don’t forget to clear the x and y variables when you are done! ENTER F44 Y X 