7.3

VOLUMES

Solids with Known Cross Sections• If A(x) is the area of a cross section of a solid

and A(x) is continuous on [a, b], then the volume of the solid from x = a to x = b is

b

a

dxxAV )(

Known Cross Sections• Ex: The base of a solid is the region enclosed by the ellipse

The cross sections are perpendicular to the x-axis and are isosceles right triangles whose hypotenuses are on the ellipse. Find the volume of the solid.

1254

22

yx

5

-5

2

-2

a

a

5

-5

2

-2

a

a

1.) Find the area of the cross section A(x).

222 )2( yaa y22 42 ya

ya 2

4

2525)(

)(

22

1)(

2

1)(

2

2

2

2

xxA

yxA

yxA

axA

2.) Set up & evaluate the integral.

2

2

2

4

2525 dx

x 3units 3

200

Unknown Cross Sections:DISC METHOD

1

Find a formula for A(x).

Sketch the solid and a typical cross section.

2

3 Find the limits of integration.

4 Integrate A(x) to find volume.

xy

A 45o wedge is cut from a cylinder of radius 3 as shown.

Find the volume of the wedge.

You could slice this wedge shape several ways, but the simplest cross section is a rectangle.

If we let h equal the height of the slice then the volume of the slice is: 2V x y h dx

Since the wedge is cut at a 45o angle:x

h45o h x

Since2 2 9x y 29y x

xy

2V x y h dx h x

29y x

22 9V x x x dx

3 2

02 9V x x dx

29u x 2 du x dx

0 9u 3 0u

10

2

9V u du

93

2

0

2

3u

227

3 18

Even though we started with a cylinder, does not enter the calculation!

Cavalieri’s Theorem:

Two solids with equal altitudes and identical parallel cross sections have the same volume.

Identical Cross Sections

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

2

x 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

2

x 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:

Math Demo

The region between the curve , and the

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

1x

y 1 4y

y x

1 1

2

3

4

1.707

2

1.577

3

1

2

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 dy

y

volume of disk

4

1

1 dy

y

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

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

2

yx

2y x

y x

2y x

2y x

2

24

0 2

yV y dy

4 2

0

1

4V y y dy

4 2

0

1

4V y y dy

42 3

0

1 1

2 12y y

168

3

8

3

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

Math Demo

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

2y x

The outer radius is:

22

yR

R

The inner radius is:

2r y

r

2y x

2

yx

2y x

y x

4 2 2

0V R r dy

2

24

02 2

2

yy dy

24

04 2 4 4

4

yy y y dy

24

04 2 4 4

4

yy y y dy

14 2 2

0

13 4

4y y y dy

432 3 2

0

3 1 8

2 12 3y y y

16 64

243 3

8

3

Find the volume of the region bounded by , , and revolved about the y-axis.

2 1y x 2x 0y 2 1y x

We can use the washer method if we split it into two parts:

25 2 2

12 1 2 1y dy

21y x 1x y

outerradius

innerradius

thicknessof slice

cylinder

5

14 1 4y dy

5

15 4y dy

52

1

15 4

2y y

25 125 5 4

2 2

25 94

2 2

164

2

8 4 12Japanese Spider CrabGeorgia Aquarium, Atlanta

If we take a vertical slice and revolve it about the y-axis

we get a cylinder.

cross section

If we add all of the cylinders together, we can reconstruct the original object.

2 1y x

Here is another way we could approach this problem:

cross section

The volume of a thin, hollow cylinder is given by:

Lateral surface area of cylinder thickness

=2 thicknessr h r is the x value of the function.

circumference height thickness

h is the y value of the function.

thickness is dx. 2=2 1 x x dx

r hthicknesscircumference

2 1y x

cross section

=2 thicknessr h

2=2 1 x x dx

r hthicknesscircumference

If we add all the cylinders from the smallest to the largest:

2 2

02 1 x x dx

2 3

02 x x dx

24 2

0

1 12

4 2x x

2 4 2

12

This is called the shell method because we use cylindrical shells.

2 1y x

Math Demo

2410 16

9y x x

Find the volume generated when this shape is revolved about the y axis.

We can’t solve for x, so we can’t use a horizontal slice directly.

2410 16

9y x x

Shell method:

Lateral surface area of cylinder

=circumference height

=2 r h Volume of thin cylinder 2 r h dx

If we take a vertical sliceand revolve it about the y-axiswe get a cylinder.

2410 16

9y x x Volume of thin cylinder 2 r h dx

8 2

2

42 10 16

9x x x dx r

h thickness

160

3502.655 cm

Note: When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.

circumference

When the strip is parallel to the axis of rotation, use the shell method.

When the strip is perpendicular to the axis of rotation, use the washer method.

• To find surface area, we can slice a solid and approximate the surface area of these slices by 2π ● f(x) ● Δs, where Δs is the slant height of the slice.

xxfs k 2))('(1

• We will see in Section 7.4 that Δs can be written as

• To find surface area, use

b

a

dxdx

dyy

2

12(SA will exist if f and

f’ are continuous on [a, b]