Приклад 1

z = x^2+y^2 \\

D=\{(x;y)|x^2+y^2 \leq 1 \} \\

I = \iint_{S} (x^2+y^2+z) dxdz = \\

z=x^2+y^2: \\

J = \begin{vmatrix} \frac{\partial x}{\partial x} & \frac{\partial x}{\partial y} \\ \frac{\partial z}{\partial x} & \frac{\partial z}{\partial y} \end{vmatrix}

= \begin{vmatrix} 1 & 0 \\ 2x & 2y \end{vmatrix} = 2y \\

I = \iint_D 2(x^2+y^2) 2y dxdy = \\

= \int_{0}^{1} \int_{-\pi}^{\pi} 4r^2 rsin\phi r dr d\phi = \\

= \int_{0}^{1} 4 r^4 (-cos \phi)|_{-\pi}^{\pi} dr = \\

= \int_{0}^{1} \frac{4r^5}{5} (1-(1)) dr = \int_{0}^{1} 0 dx = 0