Задача 72153 вычислить криволинейный интеграл...
Условие
математика ВУЗ
55
Решение
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[m]d σ =\sqrt{1+(z`_(x))^2+(z`_(y))^2}dxdy[/m]
[m]z=x[/m]
[m]d σ =\sqrt{1+(1)^2+(0)^2}dxdy=\sqrt{2}dxdy[/m]
[m] ∫ ∫ _{S}zd σ = ∫ ∫ _{D}x\sqrt{2}dxdy=[/m]
D:
0 ≤ x ≤ 1
0 ≤ y ≤ 1-x
[m]= \sqrt{2}∫^{1} _{0}( ∫^{1-x} _{0}xdy)dx= \sqrt{2}∫^{1} _{0}x\cdot (y)|^{1-x} _{0}dx= \sqrt{2}∫^{1} _{0}x\cdot (1-x)dx= \sqrt{2}∫^{1} _{0}(x-x^2)dx=[/m]
[m]= \sqrt{2}(\frac{x^2}{2}-\frac{x^3}{3})|^{1} _{0}= \sqrt{2}(\frac{1^2}{2}-\frac{1^3}{3})=\frac{ \sqrt{2}}{6}[/m]