Problems
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Prove that if $P$ and $Q$ are two arbitrary partitions of $[a,b]$, then $L(P)\leq U(Q)$.
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Prove that if $f(x)$ is continuous on $[a,b]$, then the function \[ F(x)=\int_a^x\,f(u)du \] is continuous on $[a,b]$, differentiable on $(a,b)$ and its derivative is $F\,'(x)=f(x)$ for every $x$ in $(a,b)$.
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Prove that \begin{equation} \int_a^a\,f(x)dx=0. \end{equation}
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Prove that if $f(x)$ is continuous on $[a,b]$ and we have a point $e$ such that $a\lt e\lt b$, then \begin{eqnarray}&&\int_a^b\,f(x)dx\\&=&\int_a^e\,f(x)dx+\int_e^b\,f(x)dx.\end{eqnarray}
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Show that if we are given two partitions ($P$ and $Q$) that differ by a single point so that $P\subset Q$ then $L(P)\leq L(Q)\leq\int_a^b\,f(x)dx\leq U(Q)\leq U(P)$.
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Determine the integral $\int_a^e(mx+b)\,dx$ using upper and lower sums.
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Given the partition $\{0,\,\frac{\pi}{4},\,\frac{\pi}{2},\,\frac{3\pi}{4},\,\pi\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^{\pi}\cos(x)\,dx$.
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Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.
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Given the partition $\{0,\frac{1}{16},\frac{4}{16},\frac{9}{16},1\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^1\sqrt{x}\,dx$.
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Given the partition $\{0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1\}$, find $L(P)$ and $U(P)$ for the integral $]\int_0^12x\,dx$.
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Determine the integral $\int_0^bx^2\,dx$ using upper and lower sums.
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Determine the integral $\int_a^bx\,dx$ using upper and lower sums.
