Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^{\pi}\cos(x)\,dx$.

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Lesson Parent: 
Integration III: Definite Integrals As Upper and Lower Sums
Problem: 

Given the partition $\{0,\frac{\pi}{2},\,\pi\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^{\pi}\cos(x)\,dx$.

Answer: 

$L(P)=-\frac{\pi}{2}$ and $U(P)=\frac{\pi}{2}$

SAMPLE SOLUTION

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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

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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:

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And let's recall what the cosine function looks like between $0$ and $\pi$:
cosine function from 0 to pi
In particular, we see that <Sign in to see all the formulas>, <Sign in to see all the formulas> and <Sign in to see all the formulas>.

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

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and

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This tells us that

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