**Chapter 15.3, Problem 41E**Evaluate the iterated integral by converting to polar coordinates.\[ \int_{0}^{1/2} \int_{\sqrt{3}y}^{\sqrt{1-y^2}} xy^2 \, dx \, dy \]This problem requires converting the given iterated integral from Cartesian coordinates to polar coordinates to facilitate evaluation.

To evaluate the iterated integral by converting to polar coordinates.∫012∫3y1-y2xy2 dx dy Solution: The integral is given by ∫012∫3y1-y2xy2 dx dy The limits of integration are:x varies from x=3y to x=1-y2y varies from y=0 to y=12This region of integration is shown by the shaded region yellow colorin the following figure. From the figure, The region of integration is a sector of a circle with radius r=1, between the x‐axis θ=0 and the line y=13x θ=π6.So, in polar coordinates the bounds are: 0≤θ≤π60≤r≤1And, We use the following substitutions;x=r cosθy=r sinθdx dy=r dr dθTherefore, we have, ∫012∫3y1-y2xy2 dx dy=∫0π6∫01r cosθr sinθ2r dr dθ=∫0π6∫01r4 cosθ sin2θ dr dθ=∫0π6∫01r4 drcosθ sin2θ dθ=∫0π6r5501cosθ sin2θ dθ Using, ∫rn dr=rn+1n+1=∫0π6155-055cosθ sin2θ dθ=∫0π615cosθ sin2θ dθ=15∫0π6sin2θ cosθ dθ=15sin3θ30π6 Using, ∫fnθ f'θ dθ=fn+1θn+1=115sin3θ0π6=115sin3π6-sin30=115123-03 =1120Therefore, ∫012∫3y1-y2xy2 dx dy=1120