Find Areas in Polar Coordinates

[JSGhost]

I can't seem to get the correct answer. I rechecked my calculations but no luck. Any help is appreciated. Thanks.

Homework Statement

Find the area inside the larger loop and outside the smaller loop of the limacon below.
r = sqrt(3)/2 + cos(theta)

Homework Equations

A = (integral from a to b) (1/2)*r^2 d@

Half Angle Formula:
(cos@)^2 = (1/2)(1+cos2@)

The Attempt at a Solution

A=2[ (integral from 0 to 2pi/3) [ (1/2)*(sqrt(3)/2 + cos@)^2 ]d@ - (integral from 2pi/3 to pi) [ (1/2)*(sqrt(3)/2 + cos@)^2 ]d@

= (integral from 0 to 2pi/3) [ (3/4) + sqrt(3)cos@ + (cos@)^2 ]d@ - (integral from 2pi/3 to pi) [ (3/4) + sqrt(3)cos@ + (cos@)^2 ]d@

= (integral from 0 to 2pi/3) [ (3/4) + sqrt(3)cos@ + (1/2)(1+cos2@) ]d@ - (integral from 2pi/3 to pi) [ (3/4) + sqrt(3)cos@ + (1/2)(1+cos2@) ]d@

= [(3/4)@ + sqrt(3)*sin@ + (1/2)@ + (1/4)(sin2@) ] (integral from 0 to 2pi/3) - [(3/4)@ + sqrt(3)*sin@ + (1/2)@ + (1/4)(sin2@) ] (integral from 2pi/3 to pi)

= [ (3/4)(2pi/3) + sqrt(3)*sin(2pi/3) + (1/2)(2pi/3) + (1/4)(sin( 2*(2pi/3) ) ] - (0 + 0 + 0 + 0) - [(3/4)(pi) + sqrt(3)*sin(pi) + (1/2)(pi) + (1/4)(sin( 2*(pi) ) ] + [ (3/4)(2pi/3) + sqrt(3)*sin(2pi/3) + (1/2)(2pi/3) + (1/4)(sin( 2*(2pi/3) ) ]

= ( pi/2 + 3/2 + pi/3 - sqrt(3)/8 ) - ( 3pi/4 + pi/2 ) + ( pi/2 + 3/2 + pi/3 - sqrt(3)/8 )

= 5pi/12 + 3 - sqrt(3)/4

 

[vela]

Check your limits of integration.

 

[JSGhost]

Thanks. The limits of integration was supposed to be 5pi/6 instead of 2pi/3.