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 $\{0,\,\frac{\pi}{4},\,\frac{\pi}{2},\,\frac{3\pi}{4},\,\pi\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.
$L(P)=\frac{\pi\sqrt{2}}{4}$ and $U(P)=\frac{\pi\,(2+\sqrt{2})}{4}$
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This partition breaks the closed interval <Sign in to see all the formulas> into the intervals <Sign in to see all the formulas> where
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In each of these intervals, the minimum value of <Sign in to see all the formulas> will be used to find $L(P)$ and the maximum value will be used to find $U(P)$.
Before we do the math let's find the all the <Sign in to see all the formulas>'s to make our math simpler:
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And let's recall what the sine function looks like between $0$ and $\pi$:
In particular, we see that
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Now let's find $L(P)$,
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and
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This tells us that
