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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 $\int_0^{\pi}\cos(x)\,dx$ .

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Lesson Parent:

Integration IV: Definite Integrals As Riemann Sums

Problem:

Answer:

$R(P)=\frac{\pi}{2}$

Solution:

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.

Given the partition {0,π2,π}\{0,\frac{\pi}{2},\,\pi\}, ˉx1=0\bar{x}_1=0, and ˉx2=π2\bar{x}_2=\frac{\pi}{2}, find the Riemann sum R(P)R(P) for the integral ∫π0cos(x)dx\int_0^{\pi}\cos(x)\,dx.

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SAMPLE SOLUTION

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>

GAIN AN ADVANTAGE

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 $\int_0^{\pi}\cos(x)\,dx$ .