Evaluate the following iterated integral by converting to polar coordinates. ∫8 −8∫0^(64−x^2)1/2 sin(x^2+y^2) dydx

Looks like the integral is

The integration region is the upper half of a circle in the x-y plane centered at the origin with radius 8. So, in polar coordinates, the integral is

where we substituted and , which gives

a reflection across the x-axis , followed by the by a reflection across the y-axis , and then a scale dilation of 0.5 .