Introduce a suitable change of variables:
$\iint_{S}f(ax+by+c)dxdy=2\int_{-1}^{1}\sqrt{1-u^2}f(u\sqrt{a^2+b^2 }+c)du$ where S is the unit circle and $a^2+b^2$ is nonzero
(The above excercise was taken from Apostol's Calculus vol. 2/pg. 401)
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