# 6.2 Volumes - PowerPoint PPT Presentation

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6.2 Volumes. If a region in the plane is revolved about a line, the resulting solid is called a solid of revolution , the line is called the axis of revolution . The Disc Method The Washer Method. the strip is perpendicular to the axis of revolution. A. Volume: The Disc Method.

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6.2 Volumes

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### 6.2 Volumes

If a region in the plane is revolved about a line, the resulting solid is called a solid of revolution, the line is called the axis of revolution.

• The Disc Method

• The Washer Method

the strip is perpendicular to the axis of revolution

### A. Volume: The Disc Method

Vertical Strip

Typical disc

Plane Region

r

Axis of Revolution

x=a

x= b

Volume of Typical Disc

Solid of Revolution

Volume of Solid

### B. Volume: The Washer Method

Typical Washer

Vertical Strip

Plane Region

Solid of Revolution

Axis of Revolution

a

b

Volume of Typical Washer

where

Volume of Solid

### Examples

Find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disc or washer.

Example One:

Example Two:

Solutions: Example One

Graph is a parabola with vertex of (1/4,1/2)

y

y

By disc method

1

(1/4,1/2)

r

x

x

0

Solutions: Example Two

y

y

x

x

1

3

By washer method