View
3
Download
0
Category
Preview:
Citation preview
1
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.
2
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
3
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
4
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
4
sA
5
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:
6
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 .
7
The idea can be generalized as: Volumes by Slicing
8
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
9
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
10
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
11
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.
12
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.
13
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”?
14
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.
15
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.
16
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”.
17
Volumes of solids of revolution by DISKS and WASHERS
Disk Method:
18
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).
19
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).
20
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.
21
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.
22
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
23
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
24
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.
25
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.
26
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 .
27
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 .
28
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 .
29
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
30
d. y = 3
e. 1y
31
Remark:
If the region is below the axis of rotation, be careful:
32
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.
33
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.
34
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.
35
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
36
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.
37
38
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.
39
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.
40
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.
41
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.
42
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.
43
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.
44
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.
Recommended