7.2: Volumes by Slicing – Day 2 - Washers Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 Little Rock Central High School, Little Rock, Arkansas

Bellwork– NO CALCULATOR (15 minutes): Read page 461, Example – Washer Method Applet7.2 – Washer Method Applet A)Find the area of R. B)Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the line x-axis. C)Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the line y=1. Homework: 7.2 Part 2 (15, 21, 29-31, 33, 59, 62, 63, 65, 69)

A)Find the area of R.

B) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the line x-axis.

C) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the line y=1.

The region bounded by and is revolved about the y-axis. Find the volume. The “disk” now has a hole in it, making it a “washer”. If we use a horizontal slice: The volume of the washer is: outer radius inner radius Example 1 – Volume by Washers

This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle. The washer method formulas are: and

If the same region is rotated about the line x = 2 : The outer radius is: R The inner radius is: r  Example 2 – Volume by Washers

Third Example of Washers Problem Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = x and y = x 2 Problem Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = x and y = x 2 …but about the line y = 2 instead of the x-axis …but about the line y = 2 instead of the x-axis The solid and a cross-section are illustrated on the next slide The solid and a cross-section are illustrated on the next slide

Third Example of Washers (cont’d)

Second Example of Washers (cont’d) Solution Here Solution Here So So

The formulacan be applied to any solid for which the cross- sectional area A(x) can be found The formulacan be applied to any solid for which the cross- sectional area A(x) can be found This includes solids of revolution, as shown above… This includes solids of revolution, as shown above… …but includes many other solids as well …but includes many other solids as well A Bigger Picture

The Method of Cross-Sections Intersect S with a plane P x perpendicular to the x-axis Intersect S with a plane P x perpendicular to the x-axis Call the cross- sectional area A(x) Call the cross- sectional area A(x) A(x) will vary as x increases from a to b A(x) will vary as x increases from a to b

Cross-Sections (cont’d) Divide S into “slabs” of equal width ∆x using planes at x 1, x 2,…, x n Divide S into “slabs” of equal width ∆x using planes at x 1, x 2,…, x n Like slicing a loaf of bread! Like slicing a loaf of bread! To add an infinite number of slices of bread…..we must integrate To add an infinite number of slices of bread…..we must integrate