Matematika/Gauso formulė: Skirtumas tarp puslapio versijų

:<math>V_x=\iint_S x^2 \mathbf{d}y \mathbf{d}z=\iint_S \left( \sqrt{R^2-y^2-z^2} \right)^2 \mathbf{d}y \mathbf{d}z=\iint_S \left( \sqrt{R^2-\rho^2} \right)^2 \rho \mathbf{d}\rho \mathbf{d}\phi=</math>

:<math>=\int_0^{2\pi} \left( \int_0^R \rho (R^2-\rho^2) \mathbf{d}\rho \right) \mathbf{d}\phi .</math>

:<math> \int_0^R \rho (R^2-\rho^2) \mathbf{d}\rho =\int_0^R (\rho R^2-\rho^3) \mathbf{d}\rho=(\frac{\rho^2 R^2}{2}-\frac{\rho^4}{4})|_0^R=</math>

:<math>=\frac{R^2 \cdot R^2}{2}-\frac{R^4}{4}=\frac{R^4}{4}.</math>