Question #51be1
1 Answer
Explanation:
Since the value of
This means that the cardioid
So let's find these two values:
#r_1 = r_2#
#1 - costheta = 1#
#costheta = 0#
#theta in {pi/2, (3pi)/2}#
So we're looking for the area outside of the circle but inside the cardioid, between
The area that we're looking for is equal to the area of the cardioid MINUS the area of the circle.
Now that we have our bounds and our equations, we can calculate the area. Remember the formula:
#A = int1/2r^2 d theta#
#A = int_(pi/2)^((3pi)/2)1/2(1-costheta)^2 d theta - int_(pi/2)^((3pi)/2)1/2(1)^2 d theta #
#A = int_(pi/2)^((3pi)/2)1/2(1 - 2costheta + cos^2theta) d theta - [1/2theta]_(pi/2)^((3pi)/2)#
#A = 1/2 int_(pi/2)^((3pi)/2) (1 - 2costheta + (1+cos(2theta))/2) d theta - 1/2pi#
#A = 1/2[theta - 2sintheta + theta/2 + sin(2theta)/4]_(pi/2)^((3pi)/2) - 1/2 pi#
#A = 1/2[pi - (-4) + pi/2 + 0] - 1/2pi#
#A = 1/2pi + 2 + 1/4pi - 1/2pi#
#A = 2 + pi/4#
Final Answer