Math 1432
DAY 6
Dr. Melahat Almus
[email protected]
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.
CHECK CASA FOR ONLINE QUIZ DUE DATES! No make ups.
Respect your friends. Do not distract anyone during lectures.
Bubble in PS ID and Popper Number very carefully. If you make a bubbling
mistake, your scantron can’t be saved in the system. In that case, you will not
get credit for the popper even if you turned it in.
BUBBLE IN POPPER #:
Question#
The base of a solid is the region bounded by y  e x for
0  x  ln 5 . If the cross sections perpendicular to the x-axis are squares, find the
volume of the solid.
A) 6 B) 12 C) 24
1 D) 2ln5 E) 6ln5
Dr. Almus What is a solid of revolution?
Finding the volume of a solid of revolution:
Disc Method:
Revolving about the x-axis: V 
Revolving about the y-axis: V 
2 

b
a
d
c
2
  f  x   d x
2
  g  y   dy
Dr. Almus Example: Consider the region in the first quadrant enclosed by y = 4 – x2.
Set up the integral that gives the volume of the solid formed by revolving this
region about the x-axis.
3 Dr. Almus Example: Let R be the region bounded by the y-axis and the graphs of y 
x
and y = 2. Sketch and shade the region R. Label points on the x and y-axis.
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.
4 Dr. Almus Example: Let R be the region in the first quadrant bounded by the y-axis and the
graphs of y  x 2 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.
5 Dr. Almus Example: Rotate the region enclosed by y 
sin x
0  x   about the x-
axis. Determine the volume of the solid formed.





6 Dr. Almus Example: Rotate the region enclosed by y = x2, y = 0, x = 3. about the x-axis.
Find the volume of the solid formed.









7 Dr. Almus 1
and the xx 1
axis for x   0,8 . Set up the formula that gives the volume of the solid generated
Example: Let R be the region bounded by the graph of
f  x 
by rotating R about the x-axis.
Example: Consider the region enclosed by y = x2, y = 0, x = 3. Find the volume of
the solid formed by revolving this region around the line x = 3.









8 Dr. Almus Washer Method
Revolving about the x-axis: V 
Revolving about the y-axis: V 
9 

b
a
d
c

2
2
 dx

2
2
 dy
  f  x     g  x  
  f  y     g  y  
Dr. Almus Example: Let R be the region bounded by the graphs of
f  x   x and
g ( x)  x3 . Set up the formula that gives the volume of the solid generated by
rotating R about the x-axis.
10 Dr. Almus Example: Let R be the region bounded by the graphs of
f  x   x and
g ( x)  2 x . Set up the formula that gives the volume of the solid generated by
rotating R about the y-axis.
11 Dr. Almus Example: Set up the integral(s) that give the volume of the region bounded by
y
x , x  0 , y  2 being revolved about:
a.
x – axis
c.
x=4
12 b.
y – axis
Dr. Almus d.
y=2
Exercise: Let R be the region bounded by the graph of
f  x   sinh x and
g  x   cosh x for x   0,ln10 . Setup the integral that gives the volume of the
solid generated by rotating R about the x-axis.
13 Dr. Almus Solve these exercises after class! These are pretty typical problems.
Exercise: Let
1.
2
x and y  x , from x = 0 to x=
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
2.
14  be the region bounded by y 
 be the region bounded by y  x 3
and y  8x , from x = 0 to x=
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.
Dr. Almus Solve these exercises after class! These are pretty typical problems.
Exercises:
1.
The region bounded by y  x 3 , x = 1 and the x-axis is rotated about the xaxis. Find the volume of the solid formed.
2.
The region bounded by y  co s x , x = 0, x   / 2 and the x-axis is
rotated about the x-axis. Find the volume of the solid formed.
3.
The region bounded by y  x 3 , y = 8 and the y-axis is rotated about the yaxis. Find the volume of the solid formed.
4.
The region bounded by y  x 2 , x = 1 and the x-axis is rotated about the xaxis. Find the volume of the solid formed.
5.
The region bounded by y  x 2 , y = 1 and the y-axis is rotated about the yaxis. Find the volume of the solid formed.
Check my website regularly for announcements.
www.math.uh.edu/~almus
15 Dr. Almus Given the region in the first quadrant bounded by the function
Question#
y = 4 – x2, set up the integral equation that finds the volume of the region when
rotated about y = 4, using the disk/washer method.
a.
V 
b.
V 
c.
V 
d.
V 
2
0
2
0
2
0
2
0
x
2
 4 2  dx
 x   4  dx
 4  x   4  dx
 4   x   dx
2 2
2
2 2
2
2 2
2
Given the region in the first quadrant bounded by the function y
Question#
2
= 4 – x , set up the integral equation that finds the volume of the region when
rotated about x = 2, using the disk/washer method.
4
  4  y   4  dy
a.
V 
b.
V 
c.
4
V     4  2  4  y
0 
d.
V 
16 0
4
0
2
 4   y  2   dy
2

4
0
  dy
2
 4  y  dy
2
Dr. Almus Question#
Given the region in the first quadrant bounded by the function y
2
= 4 – x , set up the integral equation that finds the volume of the region when
rotated about y = 0, using the disk/washer method.
2
4  x 
2 2
a.
V 
b.
V 
c.
V      4  x 2  dx
d.
V 
0
2
0
4  x 
dx
2 2
dx
2
2
0
2
0
 4  x  dx
2
Given the region in the first quadrant bounded by the function
Question#
2
y = 4 – x , set up the integral equation that finds the volume of the region
when rotated about x = 0, using the disk/washer method.
a.
V 
b.
V 
c.
V 
d.
V 
17 2
0
4
0
4
0
4
0
 4  y  dy
 4  y  dy
4  y dy
 0   4  y   dy
Dr. Almus