Задача 39401 подробное решение...

vk173500291

2019-09-12 20:05:00

подробное решение

sova

2019-09-13 09:31:59

x= ρ cos φ
y= ρ sin φ
dxdy= ρ d ρ d φ

0 ≤ ρ ≤ 2
0 ≤ φ ≤ [m]\frac{\pi}{2}[/m]

[m]\frac{xy}{\sqrt{x^2+y^2}}=\frac{\rho cos\phi \cdot \rho sin\phi }{\rho }[/m]

[m]\int_{0}^{2}\int^{\sqrt{4–x^2}}_{0}\frac{xy}{\sqrt{x^2+y^2}}dxdy=[/m]

[m]\int_{0}^{2}\int^{\frac{\pi }{2}}_{0}\frac{\rho cos\phi \cdot\rho sin\phi }{\rho }\rho d\rho d\phi =[/m]

[m]\int_{0}^{2}\rho^2d\rho \int_{0}^{\frac{\pi }{2}}sin\phi d(sin\phi )=\frac{\rho^3}{3}|^{2}_{0}\cdot\frac{sin^2\phi }{2}|_{0}^{\frac{\pi }{2}}=\frac{8}{3}\cdot\frac{1}{2}=\frac{4}{3}[/m]