7-2 Volume: The Disk & Washer Methods

Objective: Find volume of a solid revolution using the disk method and washer method.

Miss BattagliaAP Calculus

Rotate around the x-axis and find the volume of y=1/2 x from 0 to 4.

If a region in the plane is revolved about a line, the resulting solid is a solid of revolution, and the line is called the axis of revolution. The simplest such solid is a right circular cylinder or disk (formed by revolving a rectangle about an axis adjacent to one side of the rectangle.

Volume of disk = (area of disk)(width of disk)= πR2Δx

ΔV = πR2Δx

The Disk Method

The approximation of the volume of a solid becomes better as ||Δ|| 0 (n ∞).

Volume of a solid =

Volume of a Solid

Representative Element

Solid of revolution

To find the volume of a solid of revolution with the disk method, use one of the following:

Horizontal Axis of Revolution Vertical Axis of Revolution

The Disk Method

Find the volume of a solid formed by revolving the region bounded by the graph of

and the x-axis (0 < x < π) about the x-axis.

Using the Disk Method

Find the volume of a solid formed by revolving the region bounded by

and g(x)=1 about the line y=1.

Revolving About a Line that is Not a Coordinate Axis

1. y = - x + 1 revolved about the x-axis from 0 to 1

2. y = 4 – x2 revolved about the x-axis from 0 to 2

3. revolved about the x-axis from 1 to 4

4. revolved about the x-axis from 0 to 3

5. y = 2 and from -3 to 3

6. revolved about the x-axis from 0 to 4

7. , y = 0 from 0 to 3

8. y = 2x2, y=0, from 0 to 2

Try these!

The disk method can be extended to cover solids of revolution with holes by replacing the representative disk with a washer.

The Washer Method

Find the volume of the solid formed by revolving the region bounded by the graph of and about the x-axis.

Using the Washer Method

Find the volume of a solid formed by revolving the region bounded by the graph of y=x2+1, y=0, x=0, and x=1 about the y-axis.

Integrating with Respect to y, Two-Integral Case

A manufacturer drills a hole in the middle of a metal sphere of radius 5 in. The hole has a radius of 3 in. What is the volume of the resulting metal ring?

Manufacturing

Read 7.2 Page 465 #1-9 odd, 11, 13

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