Question #811ac

1 Answer
Nov 27, 2017

=2sqrt3-(2pi)/3

Explanation:

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r=costheta

r=1-costheta

Let's set the two functions equal to each other and solve for theta to find the points of intersection:

costheta=1-costheta

2costheta=1

costheta=1/2

theta=arccos(1/2)=pi/3 and -pi/3

The enclosed areas are between 0 and pi/3 and between 0 and -pi/3, and the two are symmetrical. We can figure out the area enclosed between 0 and pi/3 and multiply it by two:

int_0^(pi/3)(cos^2theta-(1-costheta)^2)(d(theta))

int_0^(pi/3)(cos^2theta-(1+cos^2theta-2costheta))(d(theta))

int_0^(pi/3)(cos^2theta-1-cos^2theta+2costheta)(d(theta))

=int_0^(pi/3)(2costheta-1)(d(theta))=2int_0^(pi/3)costheta(d(theta))-int_0^(pi/3)d(theta)

=(2sintheta-theta)_0^(pi/3)=2(sqrt3/2)-pi/3-2sin0-0=sqrt3-pi/3-0-0=sqrt3-pi/3

Now we multiply it by 2 to get the two enclosed areas together:

=2sqrt3-(2pi)/3