Solid of Revolution (Example: Torus)

1: DISK METHOD

2: WASHERS METHOD

6.2 Volumes by slicing (pg:421)

Volumes by disks and washers perpendicular to the x-axis (page:424)

1. Volumes by Disk Method (pg:424)

6.2.4(p. 424)

Figure 6.2.9(p. 424)

Equation (5)(p. 425)

Curve revolving about x-axis

Example 2: (pg: 425)

Find the volume of the solid that is obtained when the region under the curve

over the interval [1, 4] is revolved about the

x-axis.

xxfy )(

2: Volumes by Washer Method (pg: 426)

Washer Washers

Doughnuts are like Washers

Volumes by washers:1. Perpendicular to the x-axis2. Perpendicular to the y-axis

6.2.5(p. 425)

Figure 6.2.12(p. 425)

Equation (6)(p. 426)

Example 4: (pg: 426)

Find the volume of the solid generated when the region between the graphs of the equations

and g(x)=x over the interval [0, 2] is revolved about the x-axis.

2. Write the formula:

Solution:

1. Draw the graph:

2

2

1)( xxf

Volumes by disks and washers perpendicular to the y-axis (page:426)

Equation (8)

Figure 6.2.14 (p. 427)

Equation (7)

Example 5: (pg: 427)

Find the volume of the solid that is obtained when the region enclosed by the curve

y=2, and x=0 is revolved about the y-axis.

xxfy )(

Figure 6.2.15 (p. 427)

Volumes by washers

b

a

dxrRA )( 22

b

a

dxrRV )( 22

Vwasher = p(R2 – r2)dx

6.3.1(p. 432)

6.3.2(p. 434)

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