Polar Area Day 2 Section 10.5A Calculus BC AP/Dual, Revised ©2018 viet.dang@humbleisd.net 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

Different Types of Finding Areas Loops from Limaçon and/or another figure Inside vs Outside Area of the inside object MINUS the area of the outside object 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Steps Determine the intersection by either making the equation to zero (1 equation) or make it equal to each other Determine the symmetry and apply the equation Solve and include identities Identities 𝐬𝐢𝐧 𝟐 𝜽= 𝟏−𝐜𝐨𝐬 𝟐𝜽 𝟐 𝐜𝐨𝐬 𝟐 𝜽= 𝟏+𝐜𝐨𝐬 𝟐𝜽 𝟐 𝐬𝐢𝐧 𝟐𝜽 =𝟐 𝐬𝐢𝐧𝜽 𝐜𝐨𝐬𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 1 Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽 𝒓 𝜽 𝟑 𝟎 −𝟑 𝝅/𝟐 𝝅 𝟗 𝟑𝝅/𝟐 𝟐𝝅 𝒓 𝜽 𝟎 𝝅/𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 1 Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 1 Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 1 Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 1 Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 1 Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 1 Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 1 Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 2 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 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 2 Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 2 Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 2 Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 2 Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 2 Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 2 Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Your Turn Find the area inside the loop of 𝒓=𝟐 + 𝟒 𝐜𝐨𝐬 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

Inside and Outside Area Determine the intersection points by making both equations equal to each other Be aware of the intersection points Apply symmetry rules Use the equation: 𝟏 𝟐 𝒂 𝒃 𝑰𝒏𝒔𝒊𝒅𝒆 𝟐 − 𝑶𝒖𝒕𝒔𝒊𝒅𝒆 𝟐 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 3 Find the area inside 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 but outside the circle 𝒓=𝟐. 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 𝒓 𝜽 𝟑 𝟎 𝟏 𝝅/𝟐 −𝟏 𝝅 𝟑𝝅/𝟐 𝟐𝝅 𝒓=𝟐 𝒓 𝜽 𝟐 𝟎 𝝅/𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 3 Find the area inside 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 but outside the circle 𝒓=𝟐. 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 3 Find the area inside 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 but outside the circle 𝒓=𝟐. 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 3 Find the area inside 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 but outside the circle 𝒓=𝟐. 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 3 Find the area inside 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 but outside the circle 𝒓=𝟐. 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 3 Find the area inside 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 but outside the circle 𝒓=𝟐. 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 4 Find the area inside 𝒓=𝟑 𝐬𝐢𝐧 𝜽 and outside 𝒓=𝟐−𝐬𝐢𝐧 𝜽 𝒓=𝟑 𝐬𝐢𝐧 𝜽 𝒓 𝜽 𝟎 𝟑 𝝅/𝟐 𝝅 −𝟑 𝟑𝝅/𝟐 𝟐𝝅 𝒓=𝟐−𝐬𝐢𝐧 𝜽 𝒓 𝜽 𝟐 𝟎 𝟏 𝝅/𝟐 𝝅 𝟑 𝟑𝝅/𝟐 𝟐𝝅 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 4 Find the area inside 𝒓=𝟑 𝐬𝐢𝐧 𝜽 and outside 𝒓=𝟐−𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 4 Find the area inside 𝒓=𝟑 𝐬𝐢𝐧 𝜽 and outside 𝒓=𝟐−𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 4 Find the area inside 𝒓=𝟑 𝐬𝐢𝐧 𝜽 and outside 𝒓=𝟐−𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 4 Find the area inside 𝒓=𝟑 𝐬𝐢𝐧 𝜽 and outside 𝒓=𝟐−𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Example 4 Find the area inside 𝒓=𝟑 𝐬𝐢𝐧 𝜽 and outside 𝒓=𝟐−𝐬𝐢𝐧 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Your Turn Find the area inside 𝒓=𝟏 and outside 𝒓=𝟏−𝐜𝐨𝐬 𝜽 𝒓=𝟏 𝒓 𝜽 𝟏 𝟎 𝝅/𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 𝒓=𝟏−𝐜𝐨𝐬 𝜽 𝒓 𝜽 𝟎 𝟏 𝝅/𝟐 𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Your Turn Find the area inside 𝒓=𝟏 and outside 𝒓=𝟏−𝐜𝐨𝐬 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Your Turn Find the area inside 𝒓=𝟏 and outside 𝒓=𝟏−𝐜𝐨𝐬 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Your Turn Find the area inside 𝒓=𝟏 and outside 𝒓=𝟏−𝐜𝐨𝐬 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Your Turn Find the area inside 𝒓=𝟏 and outside 𝒓=𝟏−𝐜𝐨𝐬 𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

AP Multiple Choice Practice Question 1 (non-calculator) The area of the region inside the polar curve 𝒓= 𝟒 𝐬𝐢𝐧 𝜽 but outside the polar curve 𝒓=𝟐 𝟐 is given by (A) 𝟏 𝟐 𝝅/𝟒 𝟑𝝅/𝟒 𝟒 𝐬𝐢𝐧 𝟐 𝜽−𝟐 𝟐 𝟐 𝒅𝜽 (B) 𝟏 𝟐 𝝅/𝟒 𝟑𝝅/𝟒 𝟒 𝐬𝐢𝐧 𝜽−𝟐 𝟐 𝒅𝜽 (C) 𝟏 𝟐 𝝅/𝟒 𝟑𝝅/𝟒 𝟏𝟔 𝐬𝐢𝐧 𝟐 𝜽−𝟖 𝒅𝜽 (D) 𝟏 𝟐 𝝅/𝟒 𝟑𝝅/𝟒 𝟒 𝐬𝐢𝐧 𝟐 𝜽−𝟏 𝒅𝜽 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

AP Multiple Choice Practice Question 1 (non-calculator) The area of the region inside the polar curve 𝒓= 𝟒 𝐬𝐢𝐧 𝜽 but outside the polar curve 𝒓=𝟐 𝟐 is given by Vocabulary Connections and Process Answer 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside

§10.5A: Polar Area with Loops and Inside/Outside Assignment Worksheet 11/24/2018 11:25 AM §10.5A: Polar Area with Loops and Inside/Outside