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 $\{0,\frac{\pi}{2},\,\pi\}$, $\bar{x}_1=0$, and $\bar{x}_2=\frac{\pi}{2}$, find the Riemann sum $R(P)$ for the integral $\int_0^{\pi}\cos(x)\,dx$.
$R(P)=\frac{\pi}{2}$
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
<Sign in to see all the formulas>
Each interval has a length denoted by <Sign in to see all the formulas> and given by:
<Sign in to see all the formulas>
And let's recall what the cosine 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 $R(P)$,
<Sign in to see all the formulas>
Note that $R(P)$ for this choice of <Sign in to see all the formulas> and <Sign in to see all the formulas> is equivalent to the upper sum in this problem.
