Download presentation
Presentation is loading. Please wait.
Published byScott Hopkins Modified over 6 years ago
1
Polar Area Day 3 Section 10.5 Calculus BC AP/Dual, Revised ©2016
4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
2
Different Types of Areas
Common Interior Add both parts Limits 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
3
10.5: Polar Area: Day 3 with Common Interior
Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . 𝒓=𝟐+𝟐 𝐜𝐨𝐬𝜽 𝜽 𝒓 𝟎 𝟒 𝝅/𝟐 𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 𝒓=𝟐−𝟐 𝐜𝐨𝐬 𝜽 𝜽 𝒓 𝟎 𝝅/𝟐 𝟐 𝝅 𝟒 𝟑𝝅/𝟐 𝟐𝝅 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
4
10.5: Polar Area: Day 3 with Common Interior
Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
5
Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . Two sides of each circle “turn” Two sides of each circle “turn” 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
6
10.5: Polar Area: Day 3 with Common Interior
Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
7
10.5: Polar Area: Day 3 with Common Interior
Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
8
10.5: Polar Area: Day 3 with Common Interior
Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
9
10.5: Polar Area: Day 3 with Common Interior
Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
10
10.5: Polar Area: Day 3 with Common Interior
Example 2 Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽. 𝒓=𝟑 𝐜𝐨𝐬 𝜽 𝒓 𝜽 𝟑 𝟎 𝝅/𝟐 –𝟑 𝝅 𝟑𝝅/𝟐 𝟐𝝅 𝒓=𝟏+𝐜𝐨𝐬 𝜽 𝒓 𝜽 𝟐 𝟎 𝟏 𝝅/𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
11
10.5: Polar Area: Day 3 with Common Interior
Example 2 Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
12
10.5: Polar Area: Day 3 with Common Interior
Example 2 Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
13
10.5: Polar Area: Day 3 with Common Interior
Example 2 Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
14
10.5: Polar Area: Day 3 with Common Interior
Example 2 Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
15
10.5: Polar Area: Day 3 with Common Interior
Your Turn Find the common interior of 𝒓=𝟒 𝐬𝐢𝐧 𝜽 and 𝒓=𝟐. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
16
10.5: Polar Area: Day 2 Loops and Inside/Outside
Review Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽 𝒓 𝜽 𝟏 𝟎 𝟑 𝝅/𝟐 𝝅 −𝟏 𝟑𝝅/𝟐 𝟐𝝅 𝒓 𝜽 𝟎 𝝅/𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 4/14/2019 1:09 PM 10.5: Polar Area: Day 2 Loops and Inside/Outside
17
10.5: Polar Area: Day 2 Loops and Inside/Outside
Example 3 Setup the equation for the area between the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽 𝒓 𝜽 𝟏 𝟎 𝟑 𝝅/𝟐 𝝅 −𝟏 𝟑𝝅/𝟐 𝟐𝝅 𝒓 𝜽 𝟎 𝝅/𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 4/14/2019 1:09 PM 10.5: Polar Area: Day 2 Loops and Inside/Outside
18
10.5: Polar Area: Day 2 Loops and Inside/Outside
Example 3 Setup the equation for the area between the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽 𝒓 𝜽 𝟏 𝟎 𝟑 𝝅/𝟐 𝝅 −𝟏 𝟑𝝅/𝟐 𝟐𝝅 𝒓 𝜽 𝟎 𝝅/𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 4/14/2019 1:09 PM 10.5: Polar Area: Day 2 Loops and Inside/Outside
19
10.5: Polar Area: Day 2 Loops and Inside/Outside
Example 3 Setup the equation for the area between the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽 4/14/2019 1:09 PM 10.5: Polar Area: Day 2 Loops and Inside/Outside
20
10.5: Polar Area: Day 3 with Common Interior
Example 5 Find the area between the two spirals of 𝒓=𝜽 and 𝒓=𝟐𝜽 for 𝟎≤𝜽≤𝟐𝝅. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
21
10.5: Polar Area: Day 3 with Common Interior
Example 6 The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (calc allowed) Let 𝑹 be the region that is inside of the graph of 𝒓=𝟐 and also inside the graph of 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 , as shaded in the figure. Find the area. A particle is moving with nonzero velocity along the polar curve given by 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 has position of 𝒙 𝒕 ,𝒚 𝒕 at time 𝒕, with 𝜽=𝟎 with 𝒕=𝟎. This particle moves along the curve so that 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 . Find the value of 𝒅𝒓 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. For the particle described in part (b), 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 , find the value of 𝒅𝒚 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
22
10.5: Polar Area: Day 3 with Common Interior
Example 6a The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . Let 𝑹 be the region that is inside of the graph of 𝒓=𝟐 and also inside the graph of 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 , as shaded in the figure. Find the area. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
23
10.5: Polar Area: Day 3 with Common Interior
Example 6a The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . Let 𝑹 be the region that is inside of the graph of 𝒓=𝟐 and also inside the graph of 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 , as shaded in the figure. Find the area. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
24
10.5: Polar Area: Day 3 with Common Interior
Example 6b The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (b) A particle is moving with nonzero velocity along the polar curve given by 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 has position of 𝒙 𝒕 ,𝒚 𝒕 at time 𝒕, with 𝜽=𝟎 with 𝒕=𝟎. This particle moves along the curve so that 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 . Find the value of 𝒅𝒓 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
25
10.5: Polar Area: Day 3 with Common Interior
Example 6b The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (b) A particle is moving with nonzero velocity along the polar curve given by 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 has position of 𝒙 𝒕 ,𝒚 𝒕 at time 𝒕, with 𝜽=𝟎 with 𝒕=𝟎. This particle moves along the curve so that 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 . Find the value of 𝒅𝒓 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
26
10.5: Polar Area: Day 3 with Common Interior
Example 6c The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (c) For the particle described in part (b), 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 , find the value of 𝒅𝒚 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
27
10.5: Polar Area: Day 3 with Common Interior
Example 6c The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (c) For the particle described in part (b), 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 , find the value of 𝒅𝒚 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
28
10.5: Polar Area: Day 3 with Common Interior
Example 6c The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (c) For the particle described in part (b), 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 , find the value of 𝒅𝒚 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
29
10.5: Polar Area: Day 3 with Common Interior
Example 7 A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. Using the calculator: Find the area bounded by the curve and 𝒙-axis Find the angle 𝜽 that corresponds to the point on the curve with 𝒙-coordinate 𝒙=−𝟐. For 𝝅 𝟑 <𝜽< 𝟐𝝅 𝟑 , 𝒅𝒓 𝒅𝜽 is negative. What does your answer tell you about 𝒓? What does it tell you about the curve? At what angle 𝜽 is the interval 𝟎≤𝜽≤ 𝝅 𝟐 the curve farthest away from the origin. Justify. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
30
10.5: Polar Area: Day 3 with Common Interior
Example 7a A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. Using the calculator: Find the area bounded by the curve and x-axis 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
31
10.5: Polar Area: Day 3 with Common Interior
Example 7b A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. Using the calculator: b) Find the angle 𝜽 that corresponds to the point on the curve with 𝒙-coordinate 𝒙=−𝟐. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
32
10.5: Polar Area: Day 3 with Common Interior
Example 7c A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. For 𝝅 𝟑 <𝜽< 𝟐𝝅 𝟑 , 𝒅𝒓 𝒅𝜽 is negative. What does your answer tell you about 𝒓? What does it tell you about the curve? 𝒅𝒓 𝒅𝜽 is how fast the radius is changing with respects to rotation. Since 𝒅𝒓 𝒅𝜽 is negative it is away from the pole/origin. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
33
10.5: Polar Area: Day 3 with Common Interior
Example 7d A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. At what angle 𝜽 is the interval 𝟎≤𝜽≤ 𝝅 𝟐 the curve farthest away from the origin. Justify. 4/14/2019 1:09 PM 10.5: Polar Area: Day 3 with Common Interior
Similar presentations
© 2025 SlidePlayer.com Inc.
All rights reserved.