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 $\int_0^{\pi}\cos(x)\,dx$.
Primary tabs
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 $\int_0^{\pi}\cos(x)\,dx$.
$R(P)=-\frac{\pi}{2}$
SAMPLE SOLUTION
All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:
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 lower sum in this problem.
