Семинар 3, 23.09.2024

Задание 4а¶

Область симметрична по всем осям

\begin{align*}&\underset{[0,1]^n}{\iiint\dots\int}x_1^2+\dots+x_n^2\d x_1\d x_2\dots\d x_n=\\&\underset{[0,1]^n}{\iiint\dots\int}x^2_1\d x_1\d x_2\dots\d x_n+\dots+\underset{[0,1]^n}{\iiint\dots\int}x_n^2\d x_1\d x_2\dots\d x_n=\\ &n\underset{[0,1]^n}{\iiint\dots\int}x_1^2\d x_1\d x_2\dots\d x_n=\\ &n\underset{n-1}{\iiint\dots\int}\frac{x_1^3}{3}\biggm|^1_0\d x_2\dots\d x_n=\\ &n\underset{n-1}{\iiint\dots\int}\frac{1}{3}\d x_2\dots\d x_n=\\ &\frac{1}{3}n\underbrace{\underset{n-1}{\iiint\dots\int}1\d x_2\dots\d x_n}_{1}=\frac{1}{3}n\\ \end{align*}

\begin{align*} \underset{[0,1]^n}{\iiint\dots\int}(x_1+\ldots+x_n)^2\d x_1\d x_2\dots\d x_n \end{align*}