Change of Variables & the Jacobian · Medium problem

A sensor's response strength at distance $\sqrt{x^2+y^2}$ from its center falls off like $e^{-(x^2+y^2)}$. Find the total response $\displaystyle\iint_D e^{-(x^2+y^2)}\,dA$ collected over the disk $D$ of radius $2$ centered at the sensor.