Left-Hand Riemann Sum Help: x^3+1, b=2, a=0, and n=50

[calc67x]

I am just learning Riemann Sums and I know the basic procedure but I am stuck with large numbers.
For example, I have x^3+1 b=2 a=0 and n=50 rectangles. This is to be a left Riemann Sum.
Dx=.04
Then f(1/25)= 1.00064
f(2/25)=1.000512
This will go up for 50 intervals
Then I will add up all these numbers and multiply by .04.
I do not understand how to get the values for each f(x) all the way to 50. It would take an extremely long time.
I have a formula n^2(n+1)^2/4, and using this based on n=50 I would get 1,625,625 but I don't know how to use this to figure out the answer.
I was able to get an answer 5.84 from a calculator, but I'd like to know how to actually do this. I am studying from my book and preparing for the fall class, so right now I do not have an instructor.
Any help is appreciated!

 

[MarkFL]

I would look at the sum:

\(\displaystyle \displaystyle S_n=\frac{b-a}{n}\sum_{k=0}^{n-1}\left(f\left(a+k\frac{b-a}{n}\right)\right)\)

Now, with:

\(\displaystyle a=0,\,b=2,\,f(x)=x^3+1\)

We have:

\(\displaystyle \displaystyle S_{n}= \frac{2}{n}\sum_{k=0}^{n-1} \left(\left(\frac{2k}{n}\right)^3+1\right)= \left(\frac{2}{n}\right)^4\left(\frac{(n-1)n}{2}\right)^2+2= \left(\frac{2(n-1)}{n}\right)^2+2=\frac{4}{n^2}-\frac{8}{n}+6\)

And so:

\(\displaystyle \displaystyle S_{50}= \frac{4}{50^2}-\frac{8}{50}+6= \frac{3651}{625}=5.8416\)

Does this make sense?

 

[calc67x]

Riemann Sums

MarkFL:
Thanks this helped!