Polar graph regions - is my logic correct

I'm doing the question in the attachment. I've done part a and part b and have had a go at part c. 5pi/4 - 7root3/8 whist the answer in the book is
2pi/3 - 7root3/8.

Is my logic correct. If so I must have mucked up on the integration but if not can you point out my logic error please.

If it's not easy to read I've integrated 0.5(1-sint)^2 between pi/2 and pi/6. I've then found the area of 1/12 of a circle. Together this is the enclosed area above the OL line. I've doubled it and then added the value to the area of half the circle.

I goet pi/4 -7/16root3 from the integration

add pi/12 from circle

and then double I get the book's answer

Are you saying that my logic is correct and my integration not?

I think the logic is correct but I cannot tell where your numbers are wrong ( I cannot see all the workings)

Hi again

I redid the integration and it was the arithmetic that had gone wrong. Yippee logic correct.

Thanks