Section 10.5 – Area and Arc Length in Polar Coordinates

Area Enclosed by Polar Curves Similar to Cartesian equations, we can find the exact area of the polar region using an integral. The only exception is that we are using sectors to approximate the area, not rectangles. The area of an sector is: The area the highlighted sector is: If we integrate this area over the entire interval, it represents the total area bounded by the curve.

Area Enclosed by Polar Curves The area of the polar region of 𝑟(𝜃) from 𝜃=𝛼 and 𝜃=𝛽 is given by

Example 1 Find the area of the region in the plane enclosed by 𝑟=2(1+ cos 𝜃 ). If you created a graph, made a table, or analyzed the equation; you will see the radius sweeps out the region exactly once as 𝜃 runs from 0 to 2𝜋. (4,0) (4,2𝜋) The area is therefore:

Example 2 Find the area inside the smaller loop of 𝑟=2 cos 𝜃 +1. If you created a graph, made a table, or analyzed the equation; you will see the smaller loop is traced by the radius as 𝜃 runs from 2𝜋/3 to 4𝜋/3. (0, 2𝜋 3 ) (0, 4𝜋 3 ) The area is therefore:

Use consecutive values for the interval and triple the integral: Example 3 Find the area inside all of the loops of 𝑟= sin 3𝜃 . (0, 𝜋 3 ) If you created a graph, made a table, or analyzed the equation; you will see the size of each loop are identical. Instead of finding the whole area, you could triple the area of one loop. (0,0) Find where the curve goes through the pole: Use consecutive values for the interval and triple the integral:

Area Enclosed by Polar Curves The area of the polar region between 𝑟 𝑜 (𝜃) and 𝑟 𝑖 (𝜃) from 𝜃=𝛼 and 𝜃=𝛽 is given by It is still outside curve minus the inside curve.

Example Find the area of the region that lies inside 𝑟=1 and outside 𝑟=1− cos 𝜃 . (1, 𝜋 2 ) If you created a graph, made a table, or analyzed the equations; you will see they intersect twice. Find that intersection: The graph can confirm −𝜋/2 to 𝜋/2 is a desired interval. Other intervals like 𝜋/2 to 3𝜋/2 would give a different area. (−1, − 𝜋 2 ) On the interval, 𝑟=1 is the outside curve and 𝑟=1− cos 𝜃 is the inside curve:

NEVER ASSUME ANYTHING ABOUT A POLAR CURVE Warning! The formulas below only work if the intersection occurs at the same 𝜃 for each relation AND the interval only includes one cycle for each relation. NEVER ASSUME ANYTHING ABOUT A POLAR CURVE

Both curves do contain the point (3,0) or (3, 2𝜋 ) but… Example Find the area of the region that lies outside 𝑟=3 and inside 𝑟= 3cos 𝜃 . (−3, 𝜋 ) If you created a graph, made a table, or analyzed the equations; you will see they intersect once. Find that intersection: 3, 2𝜋 or 3,0 Both curves do contain the point (3,0) or (3, 2𝜋 ) but… Both 𝑟= 3cos 𝜃 also contains the point (−3, 𝜋 ). And this point is at the same location as (3, 2𝜋 ) . This is because 𝑟= 3cos 𝜃 completes a cycle on 0 to π but 𝑟=3 completes a cycle on 0 to 2π. The graph at right is only on 0 to π.

Example (Continued) Find the area of the region that lies outside 𝑟=3 and inside 𝑟= 3cos 𝜃 . (−3, 𝜋 ) Therefore, you can not find the area with one integral that has the same intervals. It must be broken up into two separate integrals (Remember 𝑟=3 is on the outside): 3, 2𝜋 or 3,0

Rule of Thumb for Polar Curves Never ASSUME anything about the curve. Always check the graph. Ask yourself… Do the intersections have the same angle? How many cycles are being graphed? Am I seeing the complete picture? Does my change in theta need to be decreased? Etc.