Given the partition $\{0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1\}$, find $L(P)$ and $U(P)$ for the integral $]\int_0^12x\,dx$.

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

Given the partition $\{0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1\}$, find $L(P)$ and $U(P)$ for the integral $]\int_0^12x\,dx$.

Answer: 

$L(P)=\frac{3}{4}$ and $U(P)=\frac{5}{4}$

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Solution: 

This partition breaks the closed interval $[0,1]$ 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 $U(P)$.

Therefore,

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and

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This tells us that

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