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 .

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 on the interval .

Let , and thus

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.