Single Variable Calculus, Chapter 5, 5.2, Section 5.2, Problem 30

Find an expression of $\displaystyle \int^{2 \pi}_0 x^2 \sin x dx$ as a limit of Riemann Sums. Do not evaluate the limit


$
\begin{equation}
\begin{aligned}

\Delta x =& \frac{b - a}{n}
\\
\\
\Delta x =& \frac{2 \pi - 0}{n}
\\
\\
\Delta x =& \frac{2 \pi}{n}
\\
\\
xi =& a + i \Delta x
\\
\\
xi =& 0 + \frac{2 \pi i}{n}
\\
\\
xi =& \frac{2 \pi i}{n}


\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

\int^{2 \pi}_0 f(x) dx =& \int^{2 \pi}_0 x^2 \sin x dx
\\
\\
\int^{2 \pi}_0 f(x) dx =& \lim_{n \to \infty} \sum \limits_{i = 1}^n f(xi) \Delta x
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\int^{2 \pi}_0 x^2 \sin x dx =& \lim_{n \to \infty} \sum \limits_{i = 1}^n f \left( \frac{2 \pi i}{n } \right) \left( \frac{2 \pi}{n} \right)
\\
\\
\int^{2 \pi}_0 x^2 \sin x dx =& \lim_{n \to \infty} \sum \limits_{i = 1}^n \left( \frac{2 \pi i}{n} \right)^2 \sin \left(\frac{2 \pi i}{n} \right) \left(\frac{2 \pi}{n} \right)
\\
\\
\int^{2 \pi}_0 x^2 \sin x dx =& \lim_{n \to \infty} \sum \limits_{i = 1}^n \left( \frac{4 \pi^2 i^2}{n^2} \right) \left( \frac{2 \pi}{n} \right) \sin \left( \frac{2 \pi i}{n} \right)
\\
\\
\int^{2 \pi}_0 x^2 \sin x dx =& \lim_{n \to \infty} \sum \limits_{i = 1}^n \frac{8 \pi 3 i^2}{n^3} \sin \left( \frac{2 \pi i}{n} \right)

\end{aligned}
\end{equation}
$