Let R be the region bounded by the x-axis and the graph of y=cos( x) between pi/2 and 3 pi/2 . Evaluate the integral R ( y2 sin3( x)+1)dA . Hint: Use symmetry.
I am kind of confused by the means of "use symmetry" for this problem, what I think it is
https://i.stack.imgur.com/KKNWU.gif
Is this correct? Thanks in advance
Your limits in the second integral where switched.
Note that $\cos x \le 0 $ for $ \pi /2 \le x \le 3\pi /2$
Thus $$\iint_R (y^2\sin^3(x)+1)~dA~=\int_{\pi/2}^{3\pi /2} \int _{\cos x}^{0} (y^2\sin^3(x)+1)~dydx~$$