Matematika/Gauso formulė: Skirtumas tarp puslapio versijų
Ištrintas turinys Pridėtas turinys
33 eilutė:
:<math>x^2=R^2-y^2-z^2,</math>
:<math>x=\sqrt{R^2-y^2-z^2}.</math>
:<math>V_x=\iint_S x^3 \mathbf{d}y \mathbf{d}z=\iint_S \left( \sqrt{R^2-y^2-z^2} \right)^3 \mathbf{d}y \mathbf{d}z=\iint_S \left( \sqrt{R^2-\rho^2} \right)^3 \mathbf{d}\rho \mathbf{d}\phi=</math>
:<math>=\int_0^{2\pi} \left( \int_0^R \sqrt{(R^2-\rho^2)^3} \mathbf{d}\rho \right) \mathbf{d}\phi .</math>
:Pasinaudodami [http://integrals.wolfram.com/index.jsp?expr=%28R%5E2+-+x%5E2%29%5E%283%2F2%29&random=false internetiniu integratoriumi], gauname, kad
:<math> \int_0^R \sqrt{(R^2-\rho^2)^3} \mathbf{d}\rho =\frac{1}{8}\left( \rho(5 R^2 - 2\rho^2)\sqrt{R^2-\rho^2}+3R^4 \arctan\frac{\rho}{\sqrt{R^2-\rho^2}} \right)
:Toliau pritaikydami ribas, kai <math>\rho</math> artėja į <math>R</math>, gauname:
:<math> \int_0^R \sqrt{(R^2-\rho^2)^3} \mathbf{d}\rho =\frac{1}{8}\left( \rho(5 R^2 - 2\rho^2)\sqrt{R^2-\rho^2}+3R^4 \arctan\frac{\rho}{\sqrt{R^2-\rho^2}} \right)|_0^R=</math>
==Taip pat skaitykite==
| |||