Ch. 6.2: Volume of a Solid of a Revolution: Disks andWashers

In this section, we will

I use the disk method to find the volume of a solid formed byrevolving a region about the x-axis.

I use the disk method to find the volume of a solid formed byrevolving a region about the y-axis.

I use the washer method to find the volume of a solid formedby revolving a region about the x-axis.

I use the washer method to find the volume of a solid formedby revolving a region about the y-axis.

I find the volume of a solid performed by revolving a regionabout a line parallel to a coordinate axis.

The disk method about the x-axisIf a function f is continuous and non-negative on a closed interval[a,b], then the volume V of the solid of revolution obtained byrevolving the region bounded by the graph of f , the x-axis, and thelines x = a and x = b about the x-axis is

V =

∫ b

aπ[f (x)]2

ExampleUse the disk method to find the volume of the solid of revolutiongenerated by revolving the region bounded by the graphs ofy = ex , the x-axis and the line x = −1 and x = 1 about the x-axis.

ExampleUse the disk method to verify the volume of sphere of radius r is4/3πr 3.

The disk method about the y-axisIf a function x = g(y) is continuous and non-negative on a closedinterval [c,d], then the volume V of the solid of revolutionobtained by revolving the region bounded by the graph of f , they-axis, and the lines y = c and y = d about the y-axis is

V =

∫ d

cπ[g(y)]2 dy

ExampleUse the disk method to find the volume of the solid of revolutiongenerated by revolving the region bounded by the graphs ofy = x3, the y -axis and the line y = 1 and y = 8 about the y-axis.

The washer method about the x-axisIf functions y = f (x) and y = g(x) are continuous on a closedinterval [a,b] and f (x) ≥ g(x) ≥ 0 on [a, b], then the volume V ofthe solid of revolution obtained by revolving the region bounded bythe graphs of f and g , and the lines x = a and x = b about thex-axis is

V =

∫ b

aπ{

[f (x)]2 − [g(x)]2}

dx

ExampleUse the washer method to find the volume V of the solid ofrevolution generated by revolving the region bounded by thegraphs of y = 2/x and y = 3− x , about the x-axis.

Example: Washer method, about y-axisFind the volume of the solid of revolution generated by revolvingthe region enclosed by the graphs of y = 2x and y = x2 about they-axis.

Revolving about a horizontal lineFind the volume of the solid of revolution generated by revolvingthe region enclosed by the graphs of y = 2x and y = x2 about theline y = −5.

Revolving about a vertical lineFind the volume of the solid of revolution generated by revolvingthe region enclosed by the graphs of y = 2x and y = x2 about theline x = −5.

Class ExerciseSet up the volume equations and find the volumes of the solidof revolution generated by revolving the region enclosed by thegraphs of y =

√x and y = x about

1. x-axis2. y-axis3. x = 24. y = −4