(Solved): Cartesian and polar coordinates are related by \\[ x=r \\cos \\theta, y=r \\sin \\theta, \\] where ...

Cartesian and polar coordinates are related by \\[ x=r \\cos \\theta, y=r \\sin \\theta, \\] where \\( r=\\sqrt{x^{2}+y^{2}} \\), and \\( 0 \\leq \\theta<2 \\pi \\). a) Show that the Jacobian for the change of variables \\( (x, y)=(r \\cos \\theta, r \\sin \\theta) \\) is: \\[ \\frac{\\partial(x, y)}{\\partial(r, \\theta)}=r \\] b) Use the change of variables in part a) to evaluate the integral \\[ \\int_{-1}^{1}\\left(\\int_{0}^{\\sqrt{1-x^{2}}} \\sin \\left(x^{2}+y^{2}\\right) d y\\right) d x \\] Sketch the region of integration.