Problems
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=0$, and $\bar{x}_2=\frac{\pi}{2}$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=\frac{\pi}{2}$, and $\bar{x}_2=\pi$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=\frac{\pi}{4}$, and $\bar{x}_2=\frac{3\pi}{4}$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=0$, and $\bar{x}_2=\frac{\pi}{2}$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\cos(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=\frac{\pi}{2}$, and $\bar{x}_2=\pi$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\cos(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=\frac{\pi}{4}$, and $\bar{x}_2=\frac{3\pi}{4}$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\cos(x)\,dx$.
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Prove that if $f(x)$ is integrable between $0$ and $1$, then \begin{equation} \lim_{n\to\infty}\,\frac{1}{n}\sum_{k=1}^n\,f\left(\frac{k}{n}\right)=\int_0^1f(x)\,dx. \end{equation}
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Prove that $\int_0^2f(x)\,dx=-1$ where \begin{equation} f(x)=\left\{\begin{array}{c}1,\,\mbox{if }0\leq x\lt 1\\ -2,\,\mbox{if }1\leq x\leq 2\end{array}\right. \end{equation} using Riemann sums.
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Prove that $\int_0^1f(x)\,dx=0$ where \begin{equation} f(x)=\left\{\begin{array}{c}1,\,\mbox{if }x=\frac{1}{2}\\ 0,\,\mbox{if }x\neq\frac{1}{2}\end{array}\right. \end{equation} using Riemann sums.
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Given the partition $\{0,\,\frac{\pi}{4},\,\frac{\pi}{2},\,\frac{3\pi}{4},\,\pi\}$, and points $\bar{x}_1=\frac{\pi}{8}$, $\bar{x}_2=\frac{3\pi}{8}$, $\bar{x}_3=\frac{5\pi}{8}$, and $\bar{x}_4=\frac{7\pi}{8}$; find $R(P)$ for $\int_0^{\pi}\sin(x)\,dx$.
