How do you find region bounded by circle r = 3cosΘ and the cardioid r = 1 + cosΘ?

Question

Cameron Alexander · Accepted Answer

#A= (5pi)/4#

Lets find the intersection of the curves in the first quadrant:

#3costheta=1+costheta => 2costheta=1 => costheta=1/2 => theta=pi/3#

The region is symmetric so we can find the area of the half of it:

#A= 2 (int_0^(pi/3) d theta int_0^(1+costheta) rdr + int_(pi/3)^(pi/2) d theta int_0^(3costheta) rdr)#

#A_1= 1/2int_0^(pi/3) d theta r^2 |_0^(1+costheta) = 1/2int_0^(pi/3) d theta (1+2costheta+cos^2theta)#

#A_1= 1/2 int_0^(pi/3) d theta (1+2costheta+(1+cos2theta)/2)#

#A_1= 1/2 ((3theta)/2+2sintheta+1/4sin2theta)|_0^(pi/3) =pi/4+(9sqrt3)/16#

#A_2= 1/2int_(pi/3)^(pi/2) d theta r^2 |_0^(3costheta)=#

#1/2int_(pi/3)^(pi/2) d theta (9cos^2theta) = 9/4 int_(pi/3)^(pi/2) d theta (1+cos2theta) =#

#= 9/4 (theta +1/2sin2theta) |_(pi/3)^(pi/2) = (3pi)/8-(9sqrt3)/16#

#A=2(pi/4+(9sqrt3)/16 + (3pi)/8-(9sqrt3)/16) = 2(5pi)/8 = (5pi)/4#

Charles Anctil · Answer

undefined To find the region bounded by the circle $ r = 3\cos\Theta $ and the cardioid $ r = 1 + \cos\Theta $, you need to first determine the points of intersection between the two curves. Then, you integrate the difference between the equations of the two curves over the interval where they intersect. The intersection points can be found by solving the equations simultaneously. After finding the intersection points, you integrate the difference between the larger curve and the smaller curve with respect to $ \Theta $ over the interval where they intersect. This will give you the area of the region bounded by the two curves.