Given the partition $\{0,\frac{1}{16},\frac{4}{16},\frac{9}{16},1\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^1\sqrt{x}\,dx$.

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

Given the partition $\{0,\frac{1}{16},\frac{4}{16},\frac{9}{16},1\}$, find $L(P)$ and $U(P)$ for the integral $\int_0^1\sqrt{x}\,dx$.

Answer: 

$L(P)=\frac{17}{32}$ and $U(P)=\frac{25}{32}$

SAMPLE SOLUTION

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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 $U(P)$.

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

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and

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This tells us that

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