Finding VolumesDisk/Washer/Shell

Chapter 6.2 & 6.3

February 27, 2007

Review of Area: Measuring a length.

Vertical Cut: Horizontal Cut:

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f (x)( )− g(x)( )

right

function

⎛⎝⎜

⎞⎠⎟−

leftfunction

⎛⎝⎜

⎞⎠⎟

f (y)( )− g(y)( )

top

function

⎛⎝⎜

⎞⎠⎟−

bottomfunction

⎛⎝⎜

⎞⎠⎟

Disk Method Slices are perpendicular to the axis of

rotation. Radius is a function of position on that axis. Therefore rotating about x axis gives an

integral in x; rotating about y gives an integral in y.

Find the volume of the solid generated by revolving the region defined by , y = 8, and x = 0 about the y-axis.

y =x3

Bounds?

Length?

Area?

Volume?

[0,8]

x = y3

π y3( )2

π y3`( )2dy

0

8

∫

Find the volume of the solid generated by revolving the region defined by , and y = 1, about the line y = 1

y =2 −x2

Bounds?

Length?

Area?

Volume?

[-1,1]

(2 −x2 )−1

π 1− x2( )2

π 1 − x2( )2dx

−1

1

∫

What if there is a “gap” between the axis of rotation and the function?

Solids of Revolution:

We determined that a cut perpendicular to the axis of rotation will either form a disk (region touches axis of rotation (AOR)) ora washer (there is a gap between the region and the AOR)

Revolved around the line y = 1, the region forms a disk

However when revolved around the x-axis, there is a “gap” between the region and the x-axis. (when we draw the radius, the radius intersects the region twice.)

Area of a Washer

R

R

r

Area

of

Outer

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟−

AreaofInner

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

π R2( ) − π r2( )

π R2( ) − r2( )⎡⎣ ⎤⎦

Note: Both R and r are measured from the axis of rotation.

Find the volume of the solid generated by revolving the region defined by , and y = 1, about the x-axis using planar slices perpendicular to the AOR.

y =2 −x2

Bounds?

Outside Radius?

Inside Radius?

Area?

[-1,1]

2 −x2

1

π 2 − x2( )2

− π 1( )2

Volume? π 2 − x2( )2

− π 1( )2

( )−1

1

∫ dx

Find the volume of the solid generated by revolving the region defined by , and y = 1, about the line y=-1.

y =2 −x2

Bounds?

Outside Radius?

Inside Radius?

Area?

[-1,1]

2 −x2( )− −1( )

1− −1( )

π 3 − x2( )2

− π 2( )2

Volume? π 3 − x2( )2

− π 2( )2

( )−1

1

∫ dx

Let R be the region in the x-y plane bounded by

Set up the integral for the volume obtained by rotating R about the x-axis using planar slices perpendicular to the axis of rotation.

y =4x,  y=

14,  and  x=2

Notice the gap:y =

4x,  y=

14,  and  x=2

Outside Radius ( R ):

Inside Radius ( r ):

4

x

1

4

Area: π4

x⎛⎝⎜

⎞⎠⎟

2

− π1

4⎛⎝⎜

⎞⎠⎟

2

Volume: π4

x⎛⎝⎜

⎞⎠⎟

2

− π1

4⎛⎝⎜

⎞⎠⎟

2

dx2

16

∫

Let R be the region in the x-y plane bounded by

Set up the integral for the volume of the solid obtained by rotating R about the x-axis, using planar slices perpendicular to the axis of rotation.

y =2x2   and  y=3x−1

Notice the gap:

Outside Radius ( R ):

Inside Radius ( r ):

3x−1

2x2

Area: π 3x −1( )2

− π 2x2( )2

Volume: π 3x −1( )2

− π 2x2( )2

( )dx1

2

1

∫

y =2x2   and  y=3x−1

Find the volume of the solid generated by revolving the region defined by , x = 3 and the x-axis about the x-axis.

y = x

Bounds?

Length? (radius)

Area?

Volume?

[0,3]

y = x

π x( )2

π xdx0

3

∫

Note in the disk/washer methods, the focus in on the radius (perpendicular to the axis of rotation) and the shape it forms. We can also look at a slice that is parallel to the axis of rotation.

Note in the disk/washer methods, the focus in on the radius (perpendicular to the axis of rotation) and the shape it forms. We can also look at a slice that is parallel to the axis of rotation.

2πrlengthofslice

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟Area:

2πr

Length of slice

Volume =

2πradiusfromAOR

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

lengthofslice

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

a

b

∫ dyx

⎛⎝⎜

⎞⎠⎟

Slice is PARALLEL to the AOR

Using on the interval [0,2] revolving around the x-axis using planar slices PARALLEL to the AOR, we find the volume:

y = x

Radius?

Length of slice?

Area?

Volume?

y

2 −y2

2π y( ) 2 −y2( )

2π y( ) 2 −y2( )dy0

2

∫

Back to example:Find volume of the solid generated by revolving the region about the y-axis using cylindrical slices

y =4x,  y=

14,  and  x=2

Length of slice ( h ):

Radius ( r ):

4

x−

14

x

Area: 2πx4x

−14

⎛⎝⎜

⎞⎠⎟

Volume: 2πx4x

−14

⎛⎝⎜

⎞⎠⎟dx

2

16

∫

Find the volume of the solid generated by revolving the region:about the y-axis, using cylindrical slices.

Length of slice ( h ):

Inside Radius ( r ):

3x−1−2x2

x

Area: 2πx 3x−1−2x2( )

Volume: 2πx 3x−1−2x2( )dx12

1

∫

y =2x2   and  y=3x−1

Try: Set up an integral integrating with respect

to y to find the volume of the solid of revolution obtained when the region bounded by the graphs of y = x2 and y = 0 and x = 2 is rotated around

a) the y-axis

b) the line y = 4