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 $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=\frac{\pi}{4}$, and $\bar{x}_2=\frac{3\pi}{4}$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$.
$R(P)=\frac{\pi}{\sqrt{2}}$
SAMPLE SOLUTION
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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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Each interval has a length denoted by <Sign in to see all the formulas> and given by:
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And let's recall what the sine function looks like between $0$ and $\pi$:
In particular, we see that <Sign in to see all the formulas> and <Sign in to see all the formulas>.
Now let's find $R(P)$,
