Definitie Integral by the definition

[Dorian Gray]

Greetings Mathematicians,

I have been having some difficulties with a definite integral problem by the definition. I was wondering if somebody could please look at my work. My main concern is addressing the absolute value mark/sign. I also included a picture of the steps how my professor told us to address the problems so you can see where I am coming from. I am open to any and all comments, suggestions, advice, etc. Thank you.

 

[galactus]

It's an absolute value, so you need two integrals.

$\displaystyle \int_{-1}^{0}-xdx+\int_{0}^{2}xdx$

For the first one:

$\displaystyle \Delta x=\frac{1}{n}$

By the right end point method: $\displaystyle -1+k\Delta x=-1+\frac{k}{n}$

Thus, rectangle k has area:

$\displaystyle f(x_{k})\Delta x=-(-1+\frac{k}{n})\cdot \frac{1}{n}=\frac{1}{n}-\frac{k}{n^{2}}$

So, the sum of the area is:

$\displaystyle \sum_{k=1}^{n}f(x_{k})\Delta x=\sum_{k=1}^{n}\left[\frac{1}{n}-\frac{k}{n^{2}}\right]$

$\displaystyle =1-\frac{1}{n^{2}}\cdot\frac{n(n+1)}{2}=\frac{1}{2}-\frac{1}{2n}$


Now, take the limit:

$\displaystyle \lim_{n\to \infty}\left[\frac{1}{2}-\frac{1}{2n}\right]=\frac{1}{2}$

Now, do the other one and add it with this result. You will get your 5/2.

 

[galactus]

Sorry, Dorian. Apparently, the latex is still not working or down again.

I guess click on quote and look at the code for now.

 

[Dorian Gray]

thanks!

Thank you Galactus! Although the Latex still wasn't working, I was able to decipher most of it. Your work is greatly appreciated!