Calculus II MAT 146

Integration Applications: Volumes of 3D Solids Our goal is to determine volumes of various shapes. Some of the shapes are the result of rotating a curve about an axis and other shapes are simply given as 3-dimensional objects. The general process we use in many problem situations is to determine the volume of some “typical” slice of the solid and then use integration to sum the volumes of an infinite number of such typical slices. The volume of a typical slice, in turn, can often be approximated by calculating the area of a face of the slice and multiplying that area by the thickness of the slice. Here, the curve y = f x( ) has been rotated about the x-axis from x = a to x = b. A slice of the resulting 3-dimensional solid has been sketched in. What is the area of the face of that typical slice? What is the volume of that typical slice?

The face of the typical slice is a circle. Its radius extends from the axis of rotation to the curve itself. Therefore, this typical slice must have radius equal to f x( ) , the value of the function at x.

To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let !x" 0 :

Volume = ( face area)(slice thickness)a

b

! = (area of a circle)(sub-division size)a

b

!

= !a

b

! radius( )2 dx = !a

b

! f x( )( )2dx = ! f (x)2 dx

a

b

! This technique for calculating the volume of a solid of rotation is often called the disk method because a typical slice is a disk.

x-axis (axis of rotation)

part of the curve

y = f(x)

thickness of the slice: Δx y = f(x)

radius: f(x)

x = a x = b

y = f(x) typical slice

Example #1: Determine the volume of the solid of revolution created when the region bounded by y = x2, y = 0, and x = 2 is rotated about the x-axis. Step 1: Draw a picture of the region to be rotated and a picture of the rotation image. Include an illustration of a typical slice.

Sketch the boundaries and identify the region to be rotated.

Reflect the region about the axis of rotation.

Visualize the rotation by representing a third dimension, including a sketch of a typical slice.

Step 2: Isolate a typical slice and calculate its volume.

Volume of a slice = ( face area)(slice thickness)= (area of circle)(slice thickness)

= ! radius( )2 !x

= ! x2( )2!x

= ! x4!x

Step 3: Set up a definite integral to represent the volume of the solid of rotation.

Total Volume = (volume of typical slice)0

2

!

= ( face area)(slice thickness)0

2

!

= !0

2

! radius( )2 dx = !0

2

! x2( )2dx = ! x4 dx

0

2

!

Step 4: Evaluate the definite integral.

Total Volume = ! x4 dx0

2

! = ! x5

5"

#$

%

&'(

)*0

2

=32!5

cubic units

Example #2: Using the same region as for Example 1, determine the volume of the solid of revolution created when the region is rotated about the line y = –1. Step 1: Draw a picture and illustrate a typical slice.

Sketch the boundaries and identify the region to be rotated.

Reflect the region about the axis of rotation.

Visualize the rotation by representing a third dimension, including a sketch of a typical slice.

Step 2: Isolate a typical slice and calculate its volume. Here, a typical slice is not a solid disk but looks like a washer, a disk with a hole in the middle. Illustrated here, this method is therefore called the washer method for determining the volume of a solid of revolution.

Volume of a slice = ( face area)(slice thickness)= (area of outside circle! area of inside circle)(slice thickness)

= ! outside radius( )2"#$%! ! inside radius( )

2"#

$%{ }&x

= ! R( )2"#$%! ! r( )

2"#

$%{ }&x

= ! 1+ x2( )2"

#'$%(! ! 1( )2"#

$%{ }&x

Step 3: Set up a definite integral to represent the volume of the solid of rotation.

Total Volume = (volume of typical slice)0

2

!

= ( face area)(slice thickness)0

2

!

= ! 1+ x2( )2"

#$%&'( ! 1( )2"#

%&{ }dx

0

2

!

Step 4: Evaluate the definite integral.

Total Volume = ! 1+ x2( )2!

"#$%&' ! 1( )2!"

$%{ }dx

0

2

(

= ! 1+ x2( )2'12{ }dx

0

2

(

=17615

! cubic units

Example #3: Using the same region as for Example 1, determine the volume of the solid of revolution created when the region is rotated about the y-axis. This rotation generates a bowl-like solid. We sub-divide the x-axis interval from x = 0 to x = 2 into sub-intervals of size ∆x. This creates a sequence of shells, each similar to a piece of tubing, say, from a paper towel roll. We

unwrap (flatten) each shell to get a three-dimensional solid whose volume is the product of length, width, and height. Step 1: Draw a picture and illustrate a typical shell.

Sketch the boundaries and identify the region to be rotated.

Reflect the region about the axis of rotation. Here, that’s the y-axis.

Visualize the rotation by representing a third dimension, including a sketch of a typical shell.

Step 2: Isolate a typical shell and calculate its volume. Here is a typical shell, with its length, width, and height identifed. This method is called the shell method for determining the volume of a solid of revolution.

Volume of a shell = (shell circumference)(thickness)(height)= (2!r)(!x)( f (x))

= (2! x)(!x) x2( )= (2! x) x2( )!x

Step 3: Set up a definite integral to represent the volume of the solid of rotation.

Total Volume = (volume of typical shell)0

2

!

= (shell circumference)(thickness)(height)0

2

!

= (2! x) x2( )dx0

2

!

Step 4: Evaluate the definite integral.

Total Volume = (2! x) x2( )dx0

2

!

= 2! x3( )dx0

2

!

= 2! x4

4"

#$

%

&'(

)*0

2

= 8! cubic units