Math 1432
DAY 8
Dr. Almus
[email protected]
www.math.uh.edu/~almus
If you email me, please mention the course (1432) in the subject line.
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.
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.
POPPER#
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) 1 6 B) 8 C) 34
 3
D) 16 E) 544
 15
Dr. Almus Section 7.5 Arc Length, Centroids and Surface Area
Arc Length
How do we find the arc length?
If the curve is traced by y  f  x  for a  x  b , then
b
L

2
1   f '  x   dx .
a
If the curve is traced by x  g  y  for c  y  d , then
d
L

2
1   g '  y   dy .
c
2 Dr. Almus Example: Give a formula for the length of the curve given by f  x   4  x 2 for
1  x  2 .
Example: Find the length of the curve traced by x 
3 2
 y  13/2 for 1  y  4 .
3
Dr. Almus 1
Exercise: Find the length of the curve given by f  x   x3/2  x from x  1 to
3
x  9.
Exercise: Find the length of the curve given by f  x  
1 2 1
x  ln x from x  1 to
4
2
x  2.
4 Dr. Almus Popper #
Question#
Find the length of the curve given by f  x   ln sec x from
x  0 to x   / 4 .

Hint: sec xdx  ln sec x  tan x  C
 2
B) 2  ln 1  2 
C) 1  ln 1  2 
D) ln 1  2 
A) 1  ln
E) None
5 Dr. Almus SURFACE AREA
How do you find the surface area of a solid of revolution?
6 Dr. Almus Fact: Let f be a positive, differentiable function with a continuous derivative
defined on an interval [a,b]. The area of the surface S obtained by revolving f
around the x-axis is given by:
b

2
A( S )  2 f  x  1   f '  x   dx .
a
Fact: Let F be a positive, differentiable function with a continuous derivative
defined on an interval [c,d]. The area of the surface S obtained by revolving F
around the y-axis is given by:
d

2
A( S )  2 F  y  1   F '  y   dy .
c
7 Dr. Almus Example: Let R be the region bounded by the graph of
f  x   2 x3 and the x-
axis for x   0,2 . Set up the integral that gives the surface area of the solid
generated when R is rotated about the x-axis.
Exercise: Let R be the region bounded by the graph of
f  x   4  x 2 and the x-
axis for x   2, 2 . Set up the integral that gives the surface area of the solid
generated when R is rotated about the x-axis.
8 Dr. Almus Popper #
Question#
Which of the following gives the surface area of the solid
generated by rotating the region bounded by f  x   x from x  0 to x  4 about
the x-axis?
4
x
A) 2 x 1  dx
4

0
4
B)
 2
0
x x
1
dx
4x
4
1
C) 2 x  dx
4

0
4
x
D) 2 x 2  dx
4

0
E) None
9 Dr. Almus Another application of integrals:
Finding the Centroid
Where is the centroid of this rectangle if it is made out of homogenous
material?
How do we find the centroid (or geometric center) of a region bounded by a
curve?
Fact: Let f be a positive, continuous function defined on an interval [a,b]. Let A be
 
the area of the region R bounded by f and the x-axis. The centroid x, y is given
by:
b
x
10  xf  x dx
a
A
b
and
y
2
1
 f  x   dx
2

a
A
.
Dr. Almus Fact: Let R be the region bounded by two continuous functions f and g over the
 
interval [a,b]. Let A be the area of the region R. The centroid x, y is given by:
b
x
 x  f  x   g  x dx
a
A
b
and
y
2
2
1
 f  x     g  x   dx
2

a
A
.
For ease of notation, we may use:
b

xA  x  f  x   g  x  dx
a
b
2
2
1
yA 
 f  x     g  x   dx
2

a
Example: Let R be the region bounded by the graph of
f  x   x 2 and y = 4. Find
the centroid.
11 Dr. Almus 12 Dr. Almus