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

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Lesson Parent:

Integration III: Definite Integrals As Upper and Lower Sums

Problem:

Answer:

$L(P)=\frac{\pi\sqrt{2}}{4}$ and $U(P)=\frac{\pi\,(2+\sqrt{2})}{4}$

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>

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:

<Sign in to see all the formulas>

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>

Now let's find $L(P)$ ,

<Sign in to see all the formulas>

and

<Sign in to see all the formulas>

This tells us that

Given the partition {0,π4,π2,3π4,π}\{0,\,\frac{\pi}{4},\,\frac{\pi}{2},\,\frac{3\pi}{4},\,\pi\}, find L(P)L(P) and U(P)U(P) for the integral ∫π0sin(x)dx\int_0^{\pi}\sin(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>

<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 $\{0,\,\frac{\pi}{4},\,\frac{\pi}{2},\,\frac{3\pi}{4},\,\pi\}$ , find $L(P)$ and $U(P)$ for the integral $\int_0^{\pi}\sin(x)\,dx$ .