Math 1432
DAY 7
Dr. Almus
[email protected]
www.math.uh.edu/~almus
If you email me, please mention the course (1432) in the subject line.
Check your CASA account for Quiz due dates. Don’t miss any online quizzes!
Be considerate of others in class. Respect your friends, do not distract any one
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.
Popper #
(You can solve these questions before class to save time; the
first couple of questions are from previous material.)
Question# Given the region in the first quadrant bounded by the function y = 4 – x2, 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
2
dx 2
0
2
0
 4  x  dx 2
1 Dr. Almus Question# Given the region in the first quadrant bounded by the function 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 
2
0
4
0
4
0
4
0
 4  y  dy  4  y  dy 4  y dy  0   4  y   dy 2 Dr. Almus Section 7.4 – Volumes of Solids of Revolution
Example: Let R be the region in the first quadrant bounded by f ( x)  10  x 2
and g ( x)  3 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 line y= -1.
3 Dr. Almus c) R is rotated about the y-axis.
4 Dr. Almus Example: Let R be the region bounded by the graph of
f  x   1  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.
5 Dr. Almus SHELL METHOD
In the Disc 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.
Revolving about the y-axis or a vertical axis:
V

b
a
2 P  x  h  x  dx
Revolving about the x-axis or a horizontal axis:
V
6 
d
c
2 P  y  h  y  dy
Dr. Almus P(x); P(y) : Distance from the axis of revolution to center of
revolution; radius
h(x); h(y) : Height of rectangle (big – little), (top – bottom),
(right – left)
dx; dy :
Width of rectangle
Example: Let R be the region in the first quadrant bounded by
y  4  x2 , y  0 .
Find the volume of the solid formed by rotating R about the y – axis using shell
method.
7 Dr. Almus Exercise: Let R be the region in the first quadrant bounded by
y  4  x2 , y  0 .
Find the volume of the solid formed by rotating R about the x – axis using shell
method.
8 Dr. Almus Example: Let R be the region bounded by the graph of
f  x    x 2  4 x and
g  x   x 2 . Set up the integral that gives the volume of the solid generated by
rotating R about
a) the x-axis.
b) the y-axis.
9 Dr. Almus Example: Let R be the region bounded by
y = x2 + 1, x = 0, x = 1, y = 0.
Set up the integral that gives the volume of the solid formed by rotating R about
the y – axis.
10 Dr. Almus Example: Let R be the region bounded by
y  x 2  4, y   x  6, x  0 .
Set up the integral that gives the volume of the solid formed by rotating R about
the y – axis.
11 Dr. Almus Exercise: Let R be the region bounded by the graph of
f  x   9  x 2 and
g  x   2 x 2 . Set up the integrals that give the volume of the solid generated by
rotating R about
a) the x-axis.
b) the y-axis.
12 Dr. Almus SUMMARY
Disk/Washer Method
vs
Shell Method
POPPER#
Question#
The region bounded by y  3 x 2  3 x and y = 0 is rotated about
the y-axis. Find the volume of the solid of revolution.
A) 3
4
B) D) 3
2
E) 
4
C) 
2
 13 Dr. Almus Question# Let R be the region in the first quadrant bounded by the x‐axis and the curve y = 2x – x2. The volume produced when R is revolved about the x‐
axis is A) 16
15
B) 8
3
D) 16 E) 8 C) 4
3
Question# The region R in the first quadrant is enclosed by the lines x = 0 and y = 5 and the curve y = x2 + 1. The volume of the solid generated when R is revolved about the y‐axis is A) 14 6 B) 8 C) 34
 3
D) 16 E) 544
 15
Dr. Almus