# Compute lower and upper sum for Riemann integral

rayman123

## Homework Statement

let $$f(x)=x^2$$ Calculate upper sum and lower sum on the interval $$[-2,2]$$ when n=2

## The Attempt at a Solution

since n=2 I divide the interval into
$$[-2,0]\cup[0,2]$$

then on the interval $$[-2,0]$$ the function $$f(x)=x^2$$ has the highest valute at $$x=-2, f(-2)=4=M_{0}$$ and the lowest value is at $$x=0, f(0)=0=m_{0}$$

on the interval the situation is the same $$x=0, f(0)=0=m_{1}$$(again the lowest value) ,and at $$x=2, f(2)=4=M_{1}$$(the highest value)

thus upper sum will be
$$S_{n}=M_{0}\cdot \Delta x+M_{1}\cdot \Delta x$$ where $$\Delta x=2$$
$$S_{n}=4\cdot 2+4\cdot 2=16$$
lower sum
$$s_{n}=0\cdot 2+0\cdot 2=0$$
and here I am a bit confused cause in my homework it says'' if you calculated correctly then the difference between lower sum and upper sum should be 16'' well here it would not work...where do I make mistake?

any help appreciated

Staff Emeritus
Homework Helper
$$4\cdot 2+4\cdot 2=8$$

This does not look correct...

rayman123
This does not look correct...
it should be 16, it was a typo but this still does not show me the error

Staff Emeritus
Homework Helper
Isn't the difference between the lower sum and the upper sum 16 now??

rayman123
the difference is $$s_{n}-S_{n}=-16$$

Staff Emeritus
Yeah, of course since $S_n$ is always larger than $s_n$.
But with difference, they don't literally mean $s_n-S_n$ here. Rather, they mean something like $|s_n-S_n|$.