# 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*