[tex]S {}^{1} _{0}dx \: S {}^{x^{2} } _{0}dy \: S {}^{y} _{0}(x - 2z)dz[/tex]помогите решить. Created by AL1KO. matematika-ru

tex]помогите решить...

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June 2022 1 27 Report

[tex]S {}^{1} _{0}dx \: S {}^{x^{2} } _{0}dy \: S {}^{y} _{0}(x - 2z)dz[/tex]
помогите решить

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NNNLLL54

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$\int \limits _0^1\, dx\; \int\limits^{x^2}_0\, dy\; \int\limits^{y}_0\, (x-2z)\,dz=\int\limits^1_0\, dx\; \int\limits^{x^2}_0\, dy\; \Big(xz-\frac{2z^2}{2}\Big)\Big|_0^{y}=\\\\=\int\limits^1_0\, dx\; \int\limits^{x^2}_0\, dy\; \Big (xy-y^2\Big)=\int\limits^1_0\, dx\; \int\limits^{x^2}_0\, \Big (xy-y^2\Big)\, dy=\\\\=\int\limits^1_0\, dx\; \Big(x\cdot \frac{y^2}{2}-\frac{y^3}{3}\Big)\Big|_0^{x^2}=\int\limits^1_0\, \Big(\frac{x^5}{2}-\frac{x^6}{3}\Big)\, dx=\Big(\frac{x^6}{12}-\frac{x^7}{21}\Big)\Big|_0^1=$

$=\frac{1}{12}-\frac{1}{21}=\frac{1}{28}$

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