i.1.50 Double integrals
condition: Calculate the double integral over the area \( D: \) \[ \iint_{D} \frac{\sin \sqrt{x^{2}+y^{2}}}{\sqrt{x^{2}+y^{2}}} d x d y \] where \( D: x^{2}+y^{2}=\frac{\pi^{2}}{9}, x^{2}+y^{2}=\pi^{2} \).
Double and iterated integrals over various areas of integration, conversion from Cartesian to polar coordinates, calculation of the Jacobian, changing the limits of integration, transformation of variables in the integral.