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1 Math 1432 Dr Almus

Math 1432

Dr. Melahat Almus

aalmus@uh.edu

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 and do not distract

anyone during the lecture.

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PRACTICE:

Q# Which of the following is a cardioid?

a. r = 3 – 3 cos θ

b. r = 4 + 5 sin θ

c. r = 4 + 3 cos θ

d. r = 2 sin θ

e. r = 4 cos (4θ)

Q# Which of the following is a flower?

a. r = 3 – 3 cos θ

b. r = 4 + 5 sin θ

c. r = 4 + 3 cos θ

d. r = 2 sin θ

e. r = 4 cos (4θ)

Q# Which of the following is a limaçon with a dent (dimple)?

a. r = 3 – 3 cos θ

b. r = 4 + 5 sin θ

c. r = 4 + 3 cos θ

d. r = 2 sin θ

e. r = 4 cos (4θ)

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Q# Which of the following is a limaçon with an inner loop?

a. r = 3 – 3 cos θ

b. r = 4 + 5 sin θ

c. r = 4 + 3 cos θ

d. r = 2 sin θ

e. r = 4 cos (4θ)

Q# Which of the following is a circle?

a. r = 3 – 3 cos θ

b. r = 4 + 5 sin θ

c. r = 4 + 3 cos θ

d. r = 2 sin θ

e. r = 4 cos (4θ)

Q# The polar plot of r = 2 + 2 cos θ is a

a. flower

b. line

c. cardioid

d. limaçon with loop

e. limaçon with dent (dimple)

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Q# The polar plot of r = 5 – 2 cos θ is a

a. flower

b. line

c. cardioid

d. limaçon with loop

e. limaçon with dent (dimple)

Q# The polar plot of r = 7 – 12 cos θ is a

a. flower

b. line

c. cardioid

d. limaçon with loop

e. limaçon with dent (dimple)

Q# The polar plot of r = 2 cos 5θ is a

a. flower with 5 petals

b. flower with 2 petals

c. flower with 10 petals

d. circle with radius 5

e. circle with diameter 2

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Q# The polar plot of r = 4 cos θ is a

a. circle centered at (0, 0)

b. flower with 4 petals

c. circle with radius 4, centered at (4, 0)

d. circle with radius 2, centered at (2, 0)

e. circle with radius 1, centered at (1, 0)

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Before we start Section 10.2, read the following:

RECALL: Hints on Integrals you will see in this section:

2 1sin sin cos

2 2d C

2 1cos sin cos

2 2d C

If the inside is different; use a quick u-sub:

2 1 5 1sin 5 sin 5 cos 5

5 2 2d C

2 1 3 1sin 3 sin 3 cos 3

3 2 2d C

To integrate 2( sin )a b ; expand first:

2 21 2sin 1 4sin 4sin ...d d continue

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Section 10.2 Area in Polar Coordinates

The area of a polar region is based on the area of a sector of a circle.

Area of a circle = r2

Therefore the area of a sector of a circle is the part of the circle you want times the

area of the whole circle:

Area sector = 2 21

2 2r r

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Example: a) Find the area bounded by the graph of r = 2 + 2 sin θ.

b) Find the area of the region in the 1st quadrant that is bounded by r = 2 + 2 sin θ.

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Example: Set up the formulas to find the area inside one petal of the flower

given by

r = 2 sin (3θ).

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Example: Set up the formulas to find the area inside one petal of the flower

given by

r = 4 cos (2θ).

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Example: Set up the formulas to find the area inside THE INNER LOOP of r =

1+2sin θ

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Example: Setup the formulas to find the area between the loops of r = 1 + 2 cos θ.

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Example: Give the formulas to find the area of the region that is in quadrant 4

and inside the outer loop of the polar graph

r = 1 – 2 cos (θ)

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Recall, Polar Area is found with the formula:

21

2

b

aA d

For regions between two curves:

2 2

2 1

1

2

b

aA d

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Example: Write the integral that gives the area of the region in the first quadrant

between r = 1 + cos θ and r = cos θ.

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Example: Write the integral that gives the area inside r = 3 sin θ and outside

r = 1 + sin θ.

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Example: Write the integral to find the area between r = 2 cos θ and r = 2sin θ.

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Remark: Be careful about “finding the points of intersection” when the equations

are polar. Graph each equation to see all of the points of intersection.

For example: Find the points of intersection for r = 1- cos θ

and r = 1+ cos θ.

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Exercise: Find the points of intersection for r = cos θ and r = sin θ.

Exercise: Write the integral that gives the area of the region in the first quadrant

that is inside r = sin(3θ) and outside r = sin θ.

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Exercise: Write the integral that gives the area of the region that is interior to both

r = 2- 2sin(θ) and r = 2sin θ.

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NEXT: Arc length

Finding the length of a Polar Curve (Arc Length)

Formula: 2 2

( ) 'L c d

Example: Set up the integral to find the arclength of one petal of the curve

cos 3r .

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Example: Find the length of the perimeter of the region in quadrant 1 bounded by

2sinr and 2cosr .

Exercise: Find circumference of a circle with center (a,0) and radius a using the

arclength formula.

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POPPER #

Q# Give the integral that will determine the area inside one petal of the flower

given by r = sin (3θ).

a. b.

c. d.

e. None

Q# Give the integral that will determine the area inside one petal of the flower

given by r = cos (3θ).

a. b.

c. d.

e. None

26

02 3 dsin

3

0

1sin3

2d

2

3

0

1sin3

2d

2

6

0

1sin3

2d

2

6

0cos 3 d

6

0

1cos3

2d

2

6

0

1cos3

2d

2

3

0

1cos3

2d