Self Study Opportunity to Demonstrate Mastery Volume Using Shells Self Study Opportunity to Demonstrate Mastery

Overview Use this presentation and the online resources provided to teach yourself how to find the volume of a rotated solid using the shell method. This is an optional opportunity to demonstrate mastery and earn points. If you do not do it, your grade will not be hurt. You have nothing to lose. This ODM will serve as a replacement test grade for any lower test score. To earn the points, you must take a quiz. The score from this quiz will be your potential test replacement score. This is to be an independent, self-study. You may work together but you should not assume it will be discussed in class.

First Example

Shells (a.k.a. Cylinders) Method

Back to Our First Example

Now, let’s bring it home

Recapping why we use this technique:

Another visualization of “Shells” We rotate around one axis But integrate wrt to the other Our cross-sections are now cylinders (shells) instead of disks and washers

Remember for the lateral surface area of a cylinder: Our 3D solid could be thought of as a bunch of lateral areas of cylinders smashed tightly together.

GENERAL SHELLS FORMULAS Axis of rotation is a horizontal line Equations will be x= And boundaries are y-values Axis of rotation is vertical line Equations will be y= And boundaries are x-values

Notes on the radius The axis of rotation will always go through the center of the cylinders. Radius is ALWAYS perpendicular to the axis of rotation. Radius is distance from axis of rotation to outer edge of each cylinder. This means when rotating around the y-axis the radius is x and when rotating around the x-axis the radius is y. If rotating around a vertical line that is not the y-axis, you must adjust the radius to account for this. For example, if the area in the previous example had been rotated around x = -1, then the radius would have been (x + 1), not just x

Online Resources http://mathdemos.gcsu.edu/mathdemos/shellmethod/gallery/gallery.html YouTube – Search “calculus shells” http://archives.math.utk.edu/visual.calculus/5/volumes.6/index.html http://www.wcpss.net/success-series/ap-calculus/video/calc-lesson-13.html?size=success

Example #2 Ex. Find the volume of the solid obtained by rotating about the y-axis the region bounded by the curve and the first quadrant. GSP

h r

Example 3: Find the volume of the region formed if is bounded by the x-axis, and this region is revolved around the line x = -1. Draw a picture of it… Fill in integral, with radius and height Evaluate

Practice Problem 1 Find the volume of the solid formed if the region bounded by and the x-axis is revolved around the y-axis.

Practice Problem 2 Find the volume when the region drawn is revolved around the line x=4.