Volumes – The Disk Method

Lesson 7.2

Revolving a Function

• Consider a function f(x) on the interval [a, b]

• Now consider revolvingthat segment of curve about the x axis

• What kind of functions generated these solids of revolution?

f(x)

a b

Disks

• We seek ways of usingintegrals to determine thevolume of these solids

• Consider a disk which is a slice of the solid What is the radius What is the thickness What then, is its volume?

dx

f(x)

2Volume of slice = ( )f x dx

Disks

• To find the volume of the whole solid we sum thevolumes of the disks

• Shown as a definite integral

f(x)

a b

2( )

b

a

V f x dx

Try It Out!

• Try the function y = x3 on the interval 0 < x < 2 rotated about x-axis

Revolve About Line Not a Coordinate Axis

• Consider the function y = 2x2 and the boundary lines y = 0, x = 2

• Revolve this region about the line x = 2

• We need an expression forthe radiusin terms of y

Washers• Consider the area

between two functions rotated about the axis

• Now we have a hollow solid

• We will sum the volumes of washers

• As an integral

f(x)

a b

g(x)

2 2( ) ( )

b

a

V f x g x dx

Application

• Given two functions y = x2, and y = x3

Revolve region between about x-axis

What will be the limits of

integration?

What will be the limits of

integration?

1

2 22 3

0

V x x dx

Revolving About y-Axis

• Also possible to revolve a function about the y-axis Make a disk or a washer to be horizontal

• Consider revolving a parabola about the y-axis How to represent the

radius? What is the thickness

of the disk?

Revolving About y-Axis

• Must consider curve asx = f(y) Radius = f(y) Slice is dy thick

• Volume of the solid rotatedabout y-axis

2( )

b

a

V f y dy

Flat Washer

• Determine the volume of the solid generated by the region between y = x2 and y = 4x, revolved about the y-axis Radius of inner circle?

• f(y) = y/4

Radius of outer circle?•

Limits?• 0 < y < 16

( )f y y

Cross Sections

• Consider a square at x = c with side equal to side s = f(c)

• Now let this be a thinslab with thickness Δx

• What is the volume of the slab?

• Now sum the volumes of all such slabs

c

f(x)

2

1

( )n

ii

b af x

n

ba

Cross Sections

• This suggests a limitand an integral

c

f(x)

2

1

( )n

ii

b af x

n

ba

2 2

1

lim ( ) ( )bn

ini a

b af x f x dx

n

Cross Sections

• We could do similar summations (integrals) for other shapes Triangles Semi-circles Trapezoids

c

f(x)

ba

Try It Out

• Consider the region bounded above by y = cos x below by y = sin x on the left by the y-axis

• Now let there be slices of equilateral triangles erected on each cross section perpendicular to the x-axis

• Find the volume

Assignment

• Lesson 7.2A

• Page 463

• Exercises 1 – 29 odd

• Lesson 7.2B

• Page 464

• Exercises 31 - 39 odd, 49, 53, 57