Integrals in Polar Coordinates. 13.4 Integrals in Polar Coordinates. Area in the Plane Rita Korsunsky

Integrals in Polar Coordinates Area in the Plane x y = r=f() =

Example 1 Find the area of the region enclosed by the cardioid r=2(1+cos). o x y r P(r,) =0 = 2

Example 2. Find the area inside the smaller loop of Example 2.Find the area inside the smaller loop of the limaon r=2cos+1. x y =2 3 r=2cos+1 =

Areas Between Curves x y o

Example 3 Find the area of the region that lies inside the circle r=1 and outside the cardioid r=1-cos. y o x

                   

                 

Example 6 Set up integrals in polar coordinates that can be used to find the area of (a) the blue region and (b) the green region.

(a) The area of the blue region Using symmetry,

(b) The area of the green region