i.1.72 Double integrals
Condition: Calculate the double integral over the region \(D\). \[ \begin{array}{l} \iint_{D} \frac{\sin \sqrt{x^{2}+y^{2}}}{\sqrt{x^{2}+y^{2}}} d x d y, \\ D: x^{2}+y^{2}=\frac{\pi^{2}}{9} ; x^{2}+y^{2}=\pi^{2} . \end{array}\]
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.