Chapter 14.5, Problem 31AYU

To determine

To verify: The entries in the below table:

Using Left end points	$\text{n}$	2	4	10	100
	Area	$\text{0}\text{.5}$	$\text{0}\text{.75}$	$\text{0}\text{.9}$	$\text{0}\text{.99}$
Using right end points	$\text{n}$	2	4	10	100
	Area	$\text{1}\text{.5}$	$\text{1}\text{.25}$	$\text{1}\text{.1}$	$\text{1}\text{.01}$

Answer to Problem 31AYU

Using left endpoints:

$\text{n }=2;0+0.5=0.5$

$\text{n }=4;0+0.125+0.25+0.375=0.75$

$\text{n }=10;0+0.02+0.04+0.06+\dots \mathrm{..}+0.18=\frac{10}{2}\left(0+0.18\right)=0.9$

$\text{n }=100;0+0.0002+0.0006+\dots \dots +0.0198=\frac{100}{2}\left(0+0.0198\right)=0.99$

Using right endpoints:

$\text{n }=2;0.5+1=1.5$

$\text{n }=4;0.125+0.25+0.375+0.5=1.25$

$\text{n }=10;0.02+0.04+0.06+\dots \mathrm{..}+0.18+0.20=\frac{10}{2}\left(0.02+0.20\right)=1.1$

$\text{n }=100;0.0002+0.0006+\dots \dots +0.0198+0.02=\frac{100}{2}\left(0.0002+0.02\right)=1.01$

Explanation of Solution

Using left endpoints:

$\text{n }=2;0+0.5=0.5$

$\text{n }=4;0+0.125+0.25+0.375=0.75$

$\text{n }=10;0+0.02+0.04+0.06+\dots \mathrm{..}+0.18=\frac{10}{2}\left(0+0.18\right)=0.9$

$\text{n }=100;0+0.0002+0.0006+\dots \dots +0.0198=\frac{100}{2}\left(0+0.0198\right)=0.99$

Using right endpoints:

$\text{n }=2;0.5+1=1.5$

$\text{n }=4;0.125+0.25+0.375+0.5=1.25$

$\text{n }=10;0.02+0.04+0.06+\dots \mathrm{..}+0.18+0.20=\frac{10}{2}\left(0.02+0.20\right)=1.1$

$\text{n }=100;0.0002+0.0006+\dots \dots +0.0198+0.02=\frac{100}{2}\left(0.0002+0.02\right)=1.01$