6.2Volumes of solids of revolution

Right Circular Cylinders

Volume of Right Circular Cylinders

Example 3: pg: 425

Derive the formula for the volume of a sphere of radius r.

Solid of Revolution (Example: Torus)

1: DISK METHOD 2: WASHERS

METHOD

6.2 Volumes by slicing (pg:421)

1. Volumes by Disk Method (pg:424)

6.2.4(p. 424)

Figure 6.2.9(p. 424)

Equation (5)(p. 425)

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 )(

11–18 Find the volume of the solid that results when the regionenclosed by the given curves is revolved about the x-axis. "11. y = '25 − x2, y = 312. y = 9 − x2, y = 0 13. x = 'y, x = y/414. y = sin x, y = cos x, x = 0, x = "/4[Hint: Use the identity cos 2x = cos2 x − sin2 x.]15. y = ex, y = 0, x = 0, x = ln 316. y = e−2x, y = 0, x = 0, x = 117. y =1'4 + x2, x = −2, x = 2, y = 018. y =e3x'1 + e6x, x = 0, x = 1, y = 0

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

21)( 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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