The density at each point of a 1 cm square plate is $16+r^4$ g/cm2, where $r$ is the distance in cm from the point to the centre of the plate. What is the mass of the plate?
I know that mass is the double integration of the density function. My problem is that the function of the density is in polar coordinates but the plate is a square. Should I convert the density function into Cartesians? How do I put the limits after the conversion?
Just use Cartesian coordinates supposing the square is centered at $0$. You will obtain the integral $$\int_{-1/2}^{1/2}\int_{-1/2}^{1/2} (16+(x^2+y^2)^2)dxdy,$$ which is easy to calculate.