calculus 2almus/1432_s10o5_after.pdf2 calculus 2 - dr. almus 10.5 – arc length for parametric...

1 Calculus 2 - Dr. Almus

Calculus 2

Dr. Melahat Almus

malmus@uh.edu

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.

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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

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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

.

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The Area of a Surface of Revolution

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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.

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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

malmus@uh.edu

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 .

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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

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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

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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 .

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SURFACE AREA

How do you find the surface area of a solid of revolution?

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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 .

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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

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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.

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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

.

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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

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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

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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

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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.

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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