Volumes by Disks and Washers Or, how much toilet paper fits on one of those huge rolls, anyway??

Volume by Slicing Approximating area  rectangles Volume = length x width x height Total volume =  (A x  t) Volume of a slice = Area of a slice x Thickness of a slice A tt

Volume by Slicing Total volume =  (A x  t) VOLUME =  A dt But as we let the slices get infinitely thin, Volume = lim  (A x  t)  t  0 Recall: A = area of a slice

Rotating a Function Such a rotation traces out a solid shape (in this case, we get something like half an egg)

Volume by Slices Slice r dt

Disk Formula VOLUME =  A dt

Volume by Disks (Rotation about the x – axis) r thickness x axis y axis Slice radius

Example

Homework: p. 324 #1-4 *Sketch the region first!

Example

More Volumes f(x) g(x) rotate around x axis Slice R r Area of big circle – hole

Washer Formula VOLUME =  A dt

Volumes by Washers (about the x – axis) f(x) g(x) Slice R r dt Big R little r dx

2 The application we’ve been waiting for... 1 rotate around x axis f(x) g(x)

Toilet Paper f(x) g(x) So we see that: f(x) = 2, g(x) = x only goes from 0 to 1, so we use these as the limits of integration. Now, plugging in our values for f and g:

Example Homework: p. 324 #5, 6, 8, 9 *Sketch the region first!

Example

(horizontal)

Volume by Cross Sections Recall slices: VOLUME =  A dt A = area of cross sections of the figure  need area of common shapes Use x when cross-sections are perpendicular to the x – axis. Use y when perpendicular to the y – axis.

Example base Homework: handout #55, 56

TI-84 Calculator Methods

Test – Area and Volume