Math 1432 DAY 37 Dr. Melahat Almus If you me, please mention the course (1432) in the subject line.

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1 Math 1432 DAY 37 Dr. Melahat Almus If you 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. POPPER# Q# Which of the following is a cardioid? a. r = 3 3 cos θ b. r = sin θ c. r = cos θ d. r = 2 sin θ e. r = 4 cos (4θ) 1 Math 1432 Dr Almus

2 Q# Which of the following is a flower? a. r = 3 3 cos θ b. r = sin θ c. r = 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 = sin θ c. r = cos θ d. r = 2 sin θ e. r = 4 cos (4θ) Q# Which of the following is a limaçon with an inner loop? a. r = 3 3 cos θ b. r = sin θ c. r = cos θ d. r = 2 sin θ e. r = 4 cos (4θ) 2 Math 1432 Dr Almus

3 Q# Which of the following is a circle? a. r = 3 3 cos θ b. r = sin θ c. r = cos θ d. r = 2 sin θ e. r = 4 cos (4θ) Q# The polar plot of r = 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 = 5 2 cos θ is a a. flower b. line c. cardioid d. limaçon with loop e. limaçon with dent (dimple) 3 Math 1432 Dr Almus

4 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 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) 4 Math 1432 Dr Almus

5 RECALL: Hints on Integrals you will see in this section: 2 1 sin d sincos C cos d sincos C 2 2 If the inside is different; use a quick u-sub: sin 5d sin 5cos 5C sin 3d sin 3cos 3C To integrate ( a bsin ) ; expand first: sin d 14sin 4sin d continue... 5 Math 1432 Dr Almus

6 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 = r 2 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 = r r 2 2 Find the area of the region between the origin and the polar graph of r = ρ(θ) for θ between α and β. 1 Area 2 2 r d 6 Math 1432 Dr Almus

7 Example: a) Find the area bounded by the graph of r = sin θ. 7 Math 1432 Dr Almus

8 b) Find the area of the region in the 1 st quadrant that is bounded by r = sin θ. 8 Math 1432 Dr Almus

9 Example: Set up the formulas to find the area inside one petal of the flower given by r = 2 sin (3θ). 9 Math 1432 Dr Almus

10 Example: Set up the formulas to find the area inside one petal of the flower given by r = 4 cos (2θ). 10 Math 1432 Dr Almus

11 Example: Set up the formulas to find the area inside THE INNER LOOP of r = 1+2sin θ 11 Math 1432 Dr Almus

12 Example: Setup the formulas to find the area between the loops of r = cos θ. 12 Math 1432 Dr Almus

13 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) 13 Math 1432 Dr Almus

Write the integral that gives the area of the region in the

14 Recall, Polar Area is found with the formula: A 1 b 2 d 2 a For regions between two curves: A 1 2 b a d Example. Write the integral that gives the area of the region in the first quadrant between r = 1 + cos θ and r = cos θ. 14 Math 1432 Dr Almus

15 Example: Write the integral that gives the area inside r = 3 sin θ and outside r = 1 + sin θ. 15 Math 1432 Dr Almus

16 Example: Write the integral to find the area between r = 2 cos θ and r = 2sin θ. 16 Math 1432 Dr Almus

17 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. 17 Math 1432 Dr Almus

18 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. Exercise: Write the integral that gives the area of the region that is interior to both r = 2-2sin(θ) and r = 2sin θ. 18 Math 1432 Dr Almus

Precalculus Lesson 9.2 Graphs of Polar Equations Mrs. Snow, Instructor

Precalculus Lesson 9.2 Graphs of Polar Equations Mrs. Snow, Instructor As we studied last section points may be described in polar form or rectangular form. Likewise an equation may be written using either