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Math 1432

Dr. Melahat Almus

malmus@uh.edu

http://www.math.uh.edu/~almus

Visit CASA regularly for announcements and course material!

If you email me, please mention the course (1432) in the subject line.

Please follow “classroom behavior” policies.

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Chapter 7- Applications of Integration

Section 7.4 - Volume

Now, we work on the problem of finding the volume of 3D objects using

integration. Integration is defined using Reimann sums and by taking their limit as

n goes to infinity. That is, we can interpret the integral to be a continuous sum.

Before we start, recall the formulas for finding volumes of basic geometric

shapes:

Volume of a sphere with radius r: 34

3V r

Volume of a box with dimensions l, h, w: ( )V l w h base area h

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Volume of a right cylinder with height h and radius r: 2( )V base area h r h

LATERAL surface area of such a cylinder: 2LA rh

Reason:

Volume of a right cone with height h, radius r: 21

3V r h

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We will use “disks” often: A disk is a 2D circular shape that includes the inside of

the circle. For example: 2 2 1x y describes a disk with center at (0,0) and radius

1.

The area of a disk with radius r is: 2A r

Note: Area of an equilateral triangle with one side s: 2 3

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sA

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PART 1- Finding the Volume of a solid with known cross sections:

The figure below shows a plane region and a solid formed by translating

along a line perpendicular to the plane of . Such a solid is called a right cylinder

with cross-section .

If has area A and height h , then the volume is very simple: V A h

You have seen two elementary examples of this notion in Geometry:

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To calculate the volume of a more general solid, we introduce a coordinate axes,

and examine the cross-sections of that solid perpendicular to that coordinate axis.

Here is an example where we choose the x-axis as the coordinate axis:

By ( )A x , we mean the area of the cross-section at coordinate x .

If the cross-sectional area ( )A x continuously varies with x , we can find the

volume of the solid by integrating ( )A x from x a to x b .

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The idea can be generalized as: Volumes by Slicing

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In summary:

* If the cross section is perpendicular to the x-axis and its area is a function of x,

say A(x), then the volume of the solid from a to b is given by

b

aV A x dx

* If the cross section is perpendicular to the y-axis and its area is a function of y,

say A(y), then the volume of the solid from c to d is given by

d

cV A y dy

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Example: Let R be the region bounded by

2 6 5 ( 1)( 5)f x x x x x and the x-axis.

R is the base of a solid whose cross-sections perpendicular to the x-axis are

squares.

Click on this link to get the Geogebra file:

https://www.geogebra.org/m/DaRqTDa7

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Example: Let R be the region bounded by f 2x x 4x and the x-axis.

R is the base of a solid whose cross-sections perpendicular to the x-axis are

equilateral triangles.

Click on this link to get the Geogbera file. Move the slider to see different cross

sections.

https://www.geogebra.org/m/YGkVa7wy

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Example 1: Let R be the region bounded by ( ) 2f x x , 1x and the x-axis.

If R is the base of a solid whose cross-sections perpendicular to the x-axis are

squares, find the volume of this solid.

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Example 2: Let R be the region in the first quadrant bounded by 23y x and

3y . If R is the base of a solid whose cross-sections perpendicular to the y-axis

are semicircles, compute the volume of this solid.

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Example 3: Let R be the region bounded by 2( )f x x and 2( ) 8g x x .

If R is the base of a solid whose cross-sections perpendicular to the x-axis are

squares, set up the integrals that would give the volume of this solid.

Exercise: What if the cross-sections given in Example 3 were “perpendicular to

the y-axis”?

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Example 4: Let R be the region in the first quadrant bounded by 2( ) xf x e ,

0 1x . If R is the base of a solid whose cross-sections perpendicular to the x-

axis are squares, compute the volume of this solid.

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Exercise: Consider a solid whose base is the region inside the circle

x2 + y2 = 4. If cross sections taken perpendicular to the y-axis are squares,

compute the volume of this solid.

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Part 2 - Finding the volume of a solid of revolution:

When you revolve a plane region about an axis, the solid formed is called a solid of

revolution.

If there is no gap between the axis of rotation and the region, the cross-sections

(slices) are disks and the method is called “disk method”.

If there is a gap between the axis of rotation and the region, the cross-sections are

“flat rings” or “washers”; and the method is called the “washer method”.

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Volumes of solids of revolution by DISKS and WASHERS

Disk Method:

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Disk Method:

Revolving about the x-axis: 2

fb

aV x dx

Revolving about the y-axis: 2

gd

cV y dy

Example: Let R be the region bounded by the x-axis and the graphs of y x

and x = 4. Sketch and shade the region R. Label points on the x and y-axis.

a) Give the formula for the volume of the solid generated when the region R is

rotated about the x-axis.

b) Find the volume for the solid in (a).

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Example: Let R be the region bounded by the y-axis and the graphs of 2y x

and y = 2. Sketch and shade the region R. Label points on the x and y-axis.

a) Give the formula for the volume of the solid generated when the region R is

rotated about the y-axis.

b) Find the volume for the solid in (a).

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Example: Consider the region in the first quadrant enclosed by 24y x .

Set up the integral that gives the volume of the solid formed by revolving this

region about the x-axis.

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Example: Let R be the region in the first quadrant bounded by the y-axis and the

graphs of 2y x and y = 9. Sketch and shade the region R.

Give the formula for the volume of the solid generated when the region R is rotated

about the y-axis. Find the volume for the solid.

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WASHER METHOD

When we apply the same idea (parallel cross sections) to a region that is not

containing the axis of revolution, we might get “washers” instead of disks. That is,

if there is a gap between the region and the axis of rotation, the cross-sections

might look like washers.

The dimensions of a typical washer are:

Outer radius: ( )R x

Inner radius: ( )r x

The washer’s area is: 2 2 2 2( ) ( ) ( ) ( ) ( )A x R x r x R x r x

Consequently, the volume of the solid can be found by:

2 2( ) ( ) ( )

b b

a a

V A x dx R x r x dx

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Washer Method:

Revolving about the x-axis: 2 2b

aV f x g x dx

Revolving about the y-axis: 2 2d

cV F y G y dy

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Example: Let R be the region bounded by the graphs of f x x and

3( )g x x . Set up the formula that gives the volume of the solid generated by

rotating R about the x-axis.

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Example: Let R be the region bounded by the graphs of f x x and

( ) 2g x x . Set up the formula that gives the volume of the solid generated by

rotating R about the y-axis.

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What if we rotate around a different line?

Example: Consider the region enclosed by , 1, 4y x x x and 1y . Give the

formula for the volume of the solid formed by revolving this region around the line

1y .

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Example: Consider the region enclosed by 2y x , 0y , 3x . Set up the

integral that gives the volume of the solid formed by revolving this region around

the line 3x .

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Example: Consider the region enclosed by 2y x and the x-axis for 0 1x .

Set up the integral that gives the volume of the solid formed by revolving this

region around the line 1y .

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Example: Set up the integral(s) that give the volume of the region bounded by

, 0, 2y x x y being revolved about:

a. x – axis

b. x = 4

c. y = 2

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d. y = 3

e. 1y

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Remark:

If the region is below the axis of rotation, be careful:

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Example: Let R be the region in the first quadrant bounded by 2( ) 10f x x

and ( ) 3g x x . Set up the integrals that will give the volume of the solid formed

when

a) R is rotated about the x-axis.

b) R is rotated about the y-axis.

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EXERCISE: Let R be the region bounded by the graph of

1f x x , x=0 and y = 3.

Set up the formula that gives the volume of the solid generated by rotating R about

the line y=5.

Exercise: Let R be the region bounded by the graph of sinhf x x and

coshg x x for 0,ln10x . Setup the integral that gives the volume of the

solid generated by rotating R about the x-axis.

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SHELL METHOD

Sometimes, finding the volume of a solid using disk/washer method is very

difficult, or even impossible:

To find the volume of this solid, we would need a different method.

The idea in Shell Method is to use “nested cylinders” to find the volume of a solid

of revolution.

Shell: consider a solid cylinder with radius R and height h, and cut out a cylindrical

core from it that has radius r.

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When one shell is cut and open up, it looks like this:

Length of this “box” is: 2 (distance from the line to the axis of rotation)=2 r

Height is: ( )h f x ; the width is: dx

Surface Area is: ( ) 2A x rh ;

Volume of one shell: 2 rhdx

Hence, the volume of the solid is the continuous sum: 2

b

a

V rhdx

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In the Disk Method, the rectangle of revolution is perpendicular to the axis of

revolution.

In the Shell Method, the rectangle of revolution is parallel to the axis of

revolution.

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Example: Let R be the region in the first quadrant bounded by 24 , 0y x y . Find the volume of the solid formed by rotating R about the y –

axis using shell method.

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Example: Let R be the region in the first quadrant bounded by 24 , 0y x y . Find the volume of the solid formed by rotating R about the x –

axis using shell method.

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Example: Let R be the region bounded by the graph of 2 4f x x x and

2g x x . Set up the integral that gives the volume of the solid generated by

rotating R about

a) the y-axis.

b) the x-axis.

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Example: Let R be the region bounded by 2 4, 6, 0y x y x x .

Set up the integral that gives the volume of the solid formed by rotating R about

the y – axis.

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Exercise: Let R be the region bounded by 2, ,0 1y x y x x . Set up the

integral that gives the volume of the solid formed when R is revolved about the

line x=-2.

Exercise: Let R be the region bounded by the graph of 29f x x and

22g x x . Set up the integrals that give the volume of the solid generated by

rotating R about

a) the x-axis.

b) the y-axis.

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WORKSHEET --- VOLUME

Solve these exercises after class! These are pretty typical problems. Solutions

will not be posted; if you doubt your answer, you can ask those questions

during lab.

Exercise: Let be the region bounded by y x and 2y x , from x = 0 to x= 1.

a. Find the volume of the solid formed by rotating around the x-axis.

b. Find the volume of the solid formed by rotating around the y-axis.

Exercise: Let be the region bounded by 3y x and 8y x , from x = 0 to x= 2.

a. Find the volume of the solid formed by rotating around the x-axis.

b. Find the volume of the solid formed by rotating around the y-axis.

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Solve these exercises after class! These are pretty typical problems.

Exercises:

1. The region bounded by 3y x , x = 1 and the x-axis is rotated about the x-

axis. Find the volume of the solid formed.

2. The region bounded by cosy x , x = 0, / 2x and the x-axis is rotated

about the x-axis. Find the volume of the solid formed.

3. The region bounded by 3y x , y = 8 and the y-axis is rotated about the y-

axis. Find the volume of the solid formed.

4. The region bounded by 2y x , x = 1 and the x-axis is rotated about the x-

axis. Find the volume of the solid formed.

5. The region bounded by 2y x , y = 1 and the y-axis is rotated about the y-

axis. Find the volume of the solid formed.