Question Number 219634 by Nicholas666 last updated on 29/Apr/25
Answered by MrGaster last updated on 01/May/25
$$ \\ $$\begin{document}\begin{align}
\int{x\geq 0,y\geq 1,0\leq z\leq 3-x+y}x\sin(\pi y)\,dx\,dy\,dz&=\int{0}^{\infty}\int{1}^{\infty}\int{0}^{3-x+y}x\sin(\pi y)\,dz\,dy\,dx\\
\int{0}^{3-x+y}x\sin(\pi y)\,dz&=x\sin(\pi y)(3-x+y)\\
\int{1}^{\infty}\int{0}^{\infty}x\sin(\pi y)(3-x+y)\,dx\,dy\\
\text{Consider the integration region:}3-x+y\geq 0&\Rightarrow y\geq x-3\text{,combined with}y\geq 1\\
\text{Divide the region into two parts:}\\
1. &\0\leq x\leq 4,\y\geq 1\\
2. &\x>4,\y\geq x-3\\
\text{First part:}\int{0}^{4}\int{1}^{\infty}x\sin(\pi y)(3-x+y)\,dy\,dx\\
\text{The integral with respect to}y\text{diverges,so consider the finite region:}1\leq x\leq 3,\1\leq y\leq x\\
\int{1}^{3}\int{1}^{x}x\sin(\pi y)(3-x+y)\,dy\,dx\\
\int{1}^{x}\sin(\pi y)(3-x+y)\,dy&=\frac{-3\cos(\pi x)+x-2}{\pi}\\
\int{1}^{3}x\left(\frac{-3\cos(\pi x)+x-2}{\pi}\right)\,dx&=\frac{2}{3\pi}\\
\boxed{\dfrac{2}{3\pi}}
\end{align}
\end{document}
Commented by MrGaster last updated on 01/May/25
Complete:\begin{document}
\begin{align}
\int{x\geq 0,y\geq 1,0\leq z\leq 3-x+y}x\sin(\pi y)\,dx\,dy\,dz&=\int{0}^{\infty}\int{1}^{\infty}\int{0}^{3-x+y}x\sin(\pi y)\,dz\,dy\,dx\\
\int{0}^{3-x+y}x\sin(\pi y)\,dz&=x\sin(\pi y)(3-x+y)\\
\int{1}^{\infty}\int{0}^{\infty}x\sin(\pi y)(3-x+y)\,dx\,dy\\
\text{Consider the integration region:}3-x+y\geq 0&\Rightarrow y\geq x-3\text{,combined with}y\geq 1\\
\text{Divide the region into two parts:}\\
1. &\0\leq x\leq 4,\y\geq 1\\
2. &\x>4,\y\geq x-3\\
\text{First part:}\int{0}^{4}\int{1}^{\infty}x\sin(\pi y)(3-x+y)\,dy\,dx\\
\text{The integral with respect to}y\text{diverges,so consider the finite region:}1\leq x\leq 3,\1\leq y\leq x\\
\int{1}^{3}\int{1}^{x}x\sin(\pi y)(3-x+y)\,dy\,dx\\
\int{1}^{x}\sin(\pi y)(3-x+y)\,dy&=\frac{-3\cos(\pi x)+x-2}{\pi}\\
\int{1}^{3}x\left(\frac{-3\cos(\pi x)+x-2}{\pi}\right)\,dx&=\frac{2}{3\pi}\\
\boxed{\dfrac{2}{3\pi}}
\end{align}
\end{document}
Commented by Nicholas666 last updated on 02/May/25
$${false} \\ $$