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1 Calculus 2 - Dr. Almus
Calculus 2
Dr. Melahat Almus
If you email me, please mention the course 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 and do not distract
anyone during the lecture.
2 Calculus 2 - Dr. Almus
10.5 – Arc length for Parametric Curves
Recall: Formula for finding the arc length of a curve in rectangular form:
2
( ) 1 '( )
b
a
L c f x dx (This is covered on Section 7.5 video)
Formula for finding the arc length of a curve in polar form:
2 2
( ) '( )L c r r d
Formula for finding the arc length of a curve in parametric form:
2 2
( ) ' '( )
b
a
L c x t y t dt
Example: Find the arc length of the curve 2 33 , ( ) 2 , 0 1x t t y t t t
3 Calculus 2 - Dr. Almus
Example: Give an integral which represents the length of the curve given
parametrically by 2cos ,3sins t t t for 0 t 2.
4 Calculus 2 - Dr. Almus
Example: Give an integral which represents the length of the curve given
parametrically by 2cos 3 ,3sin 4r t t t for 0 t 2.
5 Calculus 2 - Dr. Almus
Velocity and Speed (magnitude of velocity)
If the position of a particle at time t is given by
( ) ( ), ( )s t x t y t
then the velocity is given by
( ) '( ), '( )v t x t y t
and the speed is given by
speed = 2 2
' 'v t x t y t
Acceleration: ( ) ''( ), ''( )a t x t y t
See this example for the “motion” (velocity and acceleration vectors):
https://www.geogebra.org/m/uYnkfbAb
6 Calculus 2 - Dr. Almus
Example: A particle is traveling on an elliptic path in the xy-plane so that its
position at time t is given by 2 3, 3 , 2 2s t t t t t .
Find the position, velocity and speed of this particle when 1t .
7 Calculus 2 - Dr. Almus
Example: A particle is traveling on an elliptic path in the xy-plane so that its
position at time t is given by 2cos ,3sins t t t .
Give the position, velocity and speed of the particle at time 4
t
.
8 Calculus 2 - Dr. Almus
The Area of a Surface of Revolution
9 Calculus 2 - Dr. Almus
Example: Consider the curve: ( ) 2cos( ), ( ) 2sin( ); 0x t t y t t t .
Find the area of the surface formed when this curve is rotated about the x-axis.
10 Calculus 2 - Dr. Almus
Example: Consider the curve in the first quadrant determined by the curves:
( ) 3cos( ), ( ) 4sin( )x t t y t t ;
If this curve is rotated about the x-axis, set up an integral that gives the area of the
surface of revolution.
1 Dr. Almus
Calculus 2
Dr. Almus
If you email me, please mention the course 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.
2 Dr. Almus
Section 7.5 Arc Length, Surface Area and Centroids
Arc Length
How do we find the arc length?
If the curve is traced by y f x for a x b , then
2
1 '
b
a
L f x dx .
If the curve is traced by x g y for c y d , then
2
1 '
d
c
L g y dy .
3 Dr. Almus
Example: Give a formula for the length of the curve given by 24f x x for
1 2x .
2
1 '
b
a
L f x dx
4 Dr. Almus
Example: Find the length of the curve traced by 3/22
13
x y for 1 4y .
2
1 '
d
c
L g y dy
5 Dr. Almus
Exercise: Find the length of the curve given by 3/21
3f x x x from 1x to
9x .
Exercise: Find the length of the curve given by 21 1ln
4 2f x x x from 1x to
2x .
6 Dr. Almus
SURFACE AREA
How do you find the surface area of a solid of revolution?
7 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:
2
( ) 2 1 '
b
a
A S f x f x dx .
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:
2
( ) 2 1 '
d
c
A S F y F y dy .
8 Dr. Almus
Example: Let R be the region bounded by the graph of 32f x x and the x-
axis for 0,2x . Set up the integral that gives the surface area of the solid
generated when R is rotated about the x-axis.
2
( ) 2 1 '
b
a
A S f x f x dx
9 Dr. Almus
Exercise: Let R be the region bounded by the graph of 24f x x and the x-
axis for 2,2x . Set up the integral that gives the surface area of the solid
generated when R is rotated about the x-axis.
10 Dr. Almus
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
a
xf x dx
xA
and
21
2
b
a
f x dx
yA
.
11 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
a
x f x g x dx
xA
and
2 21
2
b
a
f x g x dx
yA
.
For ease of notation, we may use:
b
a
xA x f x g x dx 2 21
2
b
a
yA f x g x dx
12 Dr. Almus
Example: Let R be the region in the first quadrant bounded by the graph of
3f x x and g x x . Find the centroid.
Step 1: Find the area of the region
11 2 43
0 0
1( )
2 4 4
x xA x x dx
.
Step 2: Use formulas to find the centroid:
b
a
xA x f x g x dx
2 21
2
b
a
yA f x g x dx
13 Dr. Almus
Fact: If a region has a line of symmetry, then the centroid lies on that line.
Exercise: Let R be the region bounded by the graph of 2f x x and y = 4. Find
the centroid.
Step 1: Find the area of the region
22 32
2 2
32(4 ) 4
3 3
xA x dx x
.
Step 2: Apply the centroid formulas. (If there is symmetry and if you can guess one
of the coordinates, you can use a shortcut.)
14 Dr. Almus
Theorem 7.5.1: Pappus’s Theorem on Volumes
15 Dr. Almus
Theorem: Suppose a solid is generated by revolving a region R about any axis
such that R does not cross the axis of rotation. Then, the volume of the solid
formed is given by:
2V RA .
Here, R is the distance from the centroid of R to the axis of rotation, and A is the
area of the region.
16 Dr. Almus
Example: Let R be the region in the first quadrant bounded by the graph of
3f x x and g x x . The centroid of this region is 8 8
,15 21
C
.
Find the volume of the solid formed when this region is revolved about
a) the x-axis.
b) the y-axis.
17 Dr. Almus
Popper #
R be the region given below, with Area = 12, and centroid (5,2).
Question# If R is revolved about the x-axis, what is the volume of the
solid formed?
a) 60pi b) 24pi c) 120pi d) 48 pi e) None
Question# If R is revolved about the y-axis, what is the volume of the
solid formed?
a) 60pi b) 24pi c) 120pi d) 48 pi e) None
