Math 1432 DAY 37 Dr. Melahat Almus almus@math.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. POPPER# 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) 1 Math 1432 Dr Almus
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) 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) 2 Math 1432 Dr Almus
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) 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
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
RECALL: Hints on Integrals you will see in this section: 2 1 sin d sincos C 2 2 2 1 cos d sincos C 2 2 If the inside is different; use a quick u-sub: 2 15 1 sin 5d sin 5cos 5C 5 2 2 2 13 1 sin 3d sin 3cos 3C 3 2 2 2 To integrate ( a bsin ) ; expand first: 2 2 12sin d 14sin 4sin d continue... 5 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 = 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 = 2 1 2 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
Example: a) Find the area bounded by the graph of r = 2 + 2 sin θ. 7 Math 1432 Dr Almus
b) Find the area of the region in the 1 st quadrant that is bounded by r = 2 + 2 sin θ. 8 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). 9 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). 10 Math 1432 Dr Almus
Example: Set up the formulas to find the area inside THE INNER LOOP of r = 1+2sin θ 11 Math 1432 Dr Almus
Example: Setup the formulas to find the area between the loops of r = 1 + 2 cos θ. 12 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) 13 Math 1432 Dr Almus
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 2 2 2 1 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
Example: Write the integral that gives the area inside r = 3 sin θ and outside r = 1 + sin θ. 15 Math 1432 Dr Almus
Example: Write the integral to find the area between r = 2 cos θ and r = 2sin θ. 16 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. 17 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. 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