7.3.1 Volume by Disks and Washers

7.3.1 Volume by Disks and Washers

I. Solids of Revolution A.) Def- If a region in the plane is revolved about a line in the plane, the resulting solid is called a SOLIDOFREVOLUTION and the line is called the AXISOFREVOLUTION.

B.) Ex. - Find the volume of the solid of revolution generated by rotating the area bounded by y = 2x, x = 3, and y = 0 about the x-axis. What does this solid of revolution look like?

Now, take a slice of the volume by passing two parallel planes perpendicular to the axis of revolution.

The slice is a circular disk obtained by rotating our representative rectangle about the x-axis. r Heightor dx

An approximation of the VOLUME of the SOR would be . The actual volume of the SOR can be found by

II. Examples Using Disks A.) Ex. 1- Find the volume of the solid of revolution generated by rotating the area bounded by about the x-axis.

B.) Ex. 2 - Find the volume of the solid of revolution generated by rotating the area bounded by , x = 0, and y = 4 about the y-axis.

III. Disks – General Case: A.) Around the x-axis: B.) Around the y-axis:

IV. Washers ALL RADII MEASURED FROM THE AXIS OF REVOLUTION. r1 r2

V. Examples Using Washers A.) Ex. 3- Find the volume of the SOR generated by rotating the area bdd by about the x-axis.

B.) Ex. 4- Find the volume of the SOR generated by rotating the area bdd by about the y-axis.

VI. Washers– General Case: A.) Around the x-axis: B.) Around the y-axis:

VII. Revolving About a Line Other Than the x and y-axes. 1.) Find the volume of the SOR generated by rotating the area bdd by about the line y=5.

VIII. About x=h or y=k– General Case: A.) Around the y=k: B.) Around the x=h:

7.3.1 Volume by Disks and Washers