Ch 7.3 VolumesCalculus Graphical, Numerical, Algebraic byFinney, Demana, Waits, Kennedy

Volume of a Solid

b

a

x

x

The definition of a solid of known integrable cross section

area A from x = 0 to x = b is the integral of A from a to b,

V = A dx.

, the area of the cross section.

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

2

0

0

0

2

V sin2

11 cos 2x

2 2

sin

4 2

4

4

= x dx

= dx

2x = x +

=

=

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.

Disk Method

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

Since we will be using the disk method to rotate shapes about other lines besides the x-axis, we will not have this formula on the formula quizzes.

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.

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

Disks Example

The region between the graph of f(x) = 2 + x cos x and the x axis over the interval [-2,2] is revolved about the x-axis to generate a solid. Find the volume of the solid

Circular Cross Sections

The region between the graph of f(x) = 2 + x cos x and the x axis over the interval [-2,2] is revolved about the x-axis to generate a solid. Find the volume of the solid

Area of the cross section =

The volume of the solid is:

2A(x) = f x

2 2 2 3

-2 -2V = A(x) dx = 2 + x cos x dx = 52.429 units

End of Ch 7.3 Day 1

Ch 7.3 Day 2: Washer Method

3

2

1

-2 2

h x = -3x

g x = 3x

f x = x2

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

Like the disk method, this formula will not be on the formula quizzes. I want you to understand the formula.

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

Washer Cross Section

The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and y = sin x is revolved about the x-axis to form a solid. Find its volume.

Washer Cross Section

The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and y = sin x is revolved about the x-axis to form a solid. Find its volume.

4 4 2 2

0 0

4

0

43

0

V = A(x) dx = cos x - sin x dx

= cos 2x dx

sin 2x = = units

2 2

Volumes of Solids: End of Day 2

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2006

7.3 Day 3

The Shell Method

Japanese Spider CrabGeorgia Aquarium, AtlantaGrows to over 12 feet wide

and lives 100 years.

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

2 1y x 2x 1y2 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 12

Japanese 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

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.

Volumes Using Cylindrical Shells

The region bounded by the curve y = , the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid using cylindrical shells.

x

Volumes Using Cylindrical Shells

The region bounded by the curve y = , the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid using cylindrical shells.

x

2 2

0V = 2 y 4 - y dy

= 8

2

2

Radius = y

x = y

Shell height = 4 - y