Polar Area Day 3 Section 10.5B Calculus BC AP/Dual, Revised ©2018 viet.dang@humbleisd.net 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

Different Types of Areas Common Interior Add both parts Limits 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . 𝒓=𝟐+𝟐 𝐜𝐨𝐬⁡𝜽 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 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

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” 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 1 Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 . 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop 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 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 2 Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽. 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 2 Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽. 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 2 Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽. 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 2 Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽. 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 2 Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽. 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Your Turn Find the common interior of 𝒓=𝟒 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝒓=𝟒 𝐜𝐨𝐬 𝜽 𝒓 𝜽 𝟒 𝟎 𝝅/𝟐 −𝟒 𝝅 𝟑𝝅/𝟐 𝟐𝝅 𝒓=𝟐 𝒓 𝜽 𝟐 𝟎 𝝅/𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Your Turn Find the common interior of 𝒓=𝟒 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Your Turn Find the common interior of 𝒓=𝟒 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Your Turn Find the common interior of 𝒓=𝟒 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Your Turn Find the common interior of 𝒓=𝟒 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop 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 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Polar Sweep file 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 3 Setup the equation for the area between the inner and outer loops 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 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 3 Setup the equation for the area between the inner and outer loops 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 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Example 3 Setup the equation for the area between the inner and outer loops of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Your Turn Setup the equation and solve for the area between the loop of 𝒓 𝜽 =𝟑𝜽 𝐬𝐢𝐧 𝜽 from 𝟎,𝟐𝝅 with a calculator 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop

§10.5B: Common Interior & Inner Loop Assignment Worksheet 2/24/2019 9:50 AM §10.5B: Common Interior & Inner Loop