Sums involving functions of the form y=a^x

I have my calculus 2 midterm today, and after going through the chapters I have found the only reccuring issue I have is with functions of the form $\displaystyle y=a^x$.

If someone here could help explain how to solve these function for areas without using integral calculus I would greatly appreciate it.

I'll give an example and my steps involved:

Solve for the area under the function $\displaystyle y=e^x$ on the interval $\displaystyle [0, b]$.

Let $\displaystyle x_i=0+(b/n)i$, and thus $\displaystyle y_i=e^{(bi)/n$

$\displaystyle \displaystyle\sum_{i=1}^{n}{y_i\Delta{x}$

$\displaystyle {\displaystyle\sum_{i=1}^{n}{e^{(bi)/n}(b/n)$

$\displaystyle \displaystyle\sum_{i=1}^{n}{e^{(bi)/n}(b/n)$

$\displaystyle (b/n)\displaystyle\sum_{i=1}^{n}{e^{(bi)/n}$

$\displaystyle (b/n)\frac{e^{b/n}-e^{b+1}}{1-e^{b/n}}$

It's at this point that I become flustered. I understand there is manipulation that must occur, but can someone give me tips or show me how, with ease? Keep in mind I am not a math wizard.