And like usual, I've solved the problem below...
As I mentioned above, this problem can be solved using calculus or geometry, but I elected to take the calculus route. Click on any of the images below to make them larger!
Above we see my initial set up of the problem. I decided to take the first quadrant of the total image and find the shaded bit there. To do that, I had to use an entire unit circle.
*CORRECTION:* I made a weird mistake in the very last section of my answer. I incorrectly "simplified" by adding (3/3) to ((pi)/3) and managed to get (4(pi)/3). Thank you to Brian in the comments section for pointing it out! The first part of the line, before the boxed answer, is correct.
Did you solve this a different way? Do you have any questions about my solution? Please leave a comment! I'd love to see a geometric solution in the comments section.