Use a graphing utility to graph the polar equations. Find the area of the given region analytically. Common interior of r= 12 sin 20 and r= 6

Question

Use a graphing utility to graph the polar equations. Find the area of the given region analytically. Common interior of r= 12 sin 20 and r= 6

Answer 1

r = 6 is a circle with radius =6 and center at the origin

r= 12sin20 is the same as

r^2 =12rsin20

x^2+y^2 = 12y

x^2 +y^2-12y +36 = 36

x^2 + (y-6)^2 = 6^2

is a circle with radius =6 and center at (0,6)

they are two circles, both with radius 6.

they overlap, with common intersection

points at (3,5) and (3,-5)

the left and right sides of the overlap area are bounded by (0,0) and (0,6), their centers

integrate with an online calclator or integration tanles