How does the calculator solve definite integrals? What process do most calculators use to calculate indefinite integrals? My teacher told me that calculators are really good at doing definite integrals, but they aren't so good at finding the derivative at a certain point. This made me wonder if calculators are just iterating through every very small delta x, which would be easy for a computer to do quickly, or if they actually solve the integrals the way a human would. 

I guess it depends based on the algorithm.
Online calculators like WolframAlpha is known to have their own code to interpret and solve equations as we do. It actually shows some step-by-step solutions. On the other hand, there are methods like Simpson's Rule to evaluate definite integrals; you can find some information at Thomas Calculus. Basically the idea is to approximate functions to easier functions that we know in a small interval, then add them all.
