Please confirm area of circle r=2sin ? + 4cos ? in 1st quad

[MAC-A-TAC]

Would someone please review my work on this problem and point out any mistakes.

Thank you for your help.

P.S. I added an attachment for my last post " Please confirm area of a cycloid" and I would greatly appreciate if you would also point out my mistake(s) in that solution, as my answer didn't match yours. The original zip attachment couldn't be opened so I reposted it as a GIF file.

 

[galactus]

I will use t instead of theta.

I didn't look through all your figures to find the mistake, but it is most likely just algebra. Your integral set up is correct.

\(\displaystyle \frac{1}{2}\int_{0}^{\frac{\pi}{2}}\left[2sin(t)+4cos(t)\right]^{2}dt=\frac{5\pi+8}{2}\)

Try expanding out and get \(\displaystyle (2sin(t)+4cos(t))^{2}=12cos^{2}(t)+8sin(2t)+4\)

This should be easier to integrate.