7.3 Volumes

Disk and Washer Methods

y x Suppose I start with this curve.

My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape.

So I put a piece of wood in a lathe and turn it to a shape to match the curve.

y xHow could we find the volume of the cone?

One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes.

The volume of each flat cylinder (disk) is:

2 the thicknessr

In this case:r= the y value of the function

thickness = a small change in x = dx

2x dx

y xThe volume of each flat cylinder (disk) is:

2 the thicknessr

If we add the volumes, we get:

24

0x dx

4

0 x dx

42

02x

8

2x dx

This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk.

If the shape is rotated about the x-axis, then the formula is:

2 b

aV y dx

2 b

aV x dy A shape rotated about the y-axis would be:

The region between the curve , and the

y-axis is revolved about the y-axis. Find the volume.

1xy

1 4y

y x

1 12

3

4

1 .7072

1 .5773

12

We use a horizontal disk.

dy

The thickness is dy.

The radius is the x value of the function .1

y

24

1

1 V dyy

volume of disk

4

1

1 dyy

4

1ln y ln 4 ln1

02ln 2 2 ln 2

The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis:

2.000574 .439 185x y y x

y

500 ft

500 22

0.000574 .439 185 y y dy

The volume can be calculated using the disk method with a horizontal disk.

324,700,000 ft

Example: Using calculus, derive the formula for finding the volume of a sphere of radius r.

The region bounded by a semicircle and its diameter shown below is revolved about the x-axis, which gives us a sphere of radius r.

dx

Area of each cross section? (circle)2yA What is y?

r–r

Equation of semicircle:222 ryx 22 xry

222 xrA

22 xrA

Example: Using calculus, derive the formula for finding the volume of a sphere of radius r.

Area of each cross section? (circle)

r–r

22 xrA

r

r

dxxrV )( 22

Volume:

(Remember r is just a number!!!)

r

r

xxrV

32

31

3232 )(

31)(

31 rrrrrrV

Example: Using calculus, derive the formula for finding the volume of a sphere of radius r.

r–r

Volume:

3232 )(

31)(

31 rrrrrrV

3333

31

31 rrrrV

33

32

32 rrV 3

34 r

Example: Find the volume of the solid generated when the region bounded by y = x2, x = 2, and y = 0 is rotated about the line x = 2.

dy

Area of each cross section? (circle)2rA What is r?

r

2

r = 2 – x 22 xA Remember, we are using a dy here!!!So,

xy

xy

2

22 yA

Example: Find the volume of the solid generated when the region bounded by y = x2, x = 2, and y = 0 is rotated about the line x = 2.

Area of each cross section? (circle)

r

2

22 yA Volume:

dyyV )2( 21

Bounds?From 0 to intersection(y-value!!!)

4

0

)2( 21 dyyV

38

The region bounded by and is revolved about the y-axis.Find the volume.

2y x 2y x

The “disk” now has a hole in it, making it a “washer”.

If we use a horizontal slice:

The volume of the washer is: 2 2 thicknessR r

2 2R r dy

outerradius

innerradius

2y x

2y x

2y x

y x

2y x

2y x

2

24

0 2yV y dy

4 2

0

14

V y y dy

4 2

0

1 4

V y y dy 4

2 3

0

1 12 12

y y

1683

83

This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle.

The washer method formula is: 2 2 b

aV R r dx

2y xIf the same region is rotated about the line x=2:

2y x

The outer radius is:

22yR

R

The inner radius is:

2r y

r

2y x

2y x

2y x

y x

4 2 2

0V R r dy

2

24

02 2

2y y dy

24

04 2 4 4

4yy y y dy

24

04 2 4 4

4yy y y dy

14 2 20

13 4 4

y y y dy 43

2 3 2

0

3 1 82 12 3

y y y

16 64243 3

83