dr. melahat almus [email protected] if you email me, please ... › ~almus › 1432_s10o2_after.pdf ·...
TRANSCRIPT
1 Math 1432 Dr Almus
Math 1432
Dr. Melahat 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 and do not distract
anyone during the lecture.
2 Math 1432 Dr Almus
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)
3 Math 1432 Dr Almus
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)
4 Math 1432 Dr Almus
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
5 Math 1432 Dr Almus
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)
6 Math 1432 Dr Almus
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
7 Math 1432 Dr Almus
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
8 Math 1432 Dr Almus
9 Math 1432 Dr Almus
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.
10 Math 1432 Dr Almus
Example: Set up the formulas to find the area inside one petal of the flower
given by
r = 2 sin (3θ).
11 Math 1432 Dr Almus
Example: Set up the formulas to find the area inside one petal of the flower
given by
r = 4 cos (2θ).
12 Math 1432 Dr Almus
Example: Set up the formulas to find the area inside THE INNER LOOP of r =
1+2sin θ
13 Math 1432 Dr Almus
Example: Setup the formulas to find the area between the loops of r = 1 + 2 cos θ.
14 Math 1432 Dr Almus
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)
15 Math 1432 Dr Almus
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
16 Math 1432 Dr Almus
Example: Write the integral that gives the area of the region in the first quadrant
between r = 1 + cos θ and r = cos θ.
17 Math 1432 Dr Almus
Example: Write the integral that gives the area inside r = 3 sin θ and outside
r = 1 + sin θ.
18 Math 1432 Dr Almus
Example: Write the integral to find the area between r = 2 cos θ and r = 2sin θ.
19 Math 1432 Dr Almus
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.
20 Math 1432 Dr Almus
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.
21 Math 1432 Dr Almus
Exercise: Write the integral that gives the area of the region that is interior to both
r = 2- 2sin(θ) and r = 2sin θ.
22 Math 1432 Dr Almus
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 .
23 Math 1432 Dr Almus
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.
24 Math 1432 Dr Almus
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