Today, we looked at functions defined by integrals in my AP class. I always find this an incredibly difficult topic to teach, so I made a huge packet this year to guide us through our classroom discussion. It helped me organize my thoughts about this tricky topic. One thing that I stumbled upon last year was a good way to conceptualize the fact that the derivative of an integral is the function itself. Basically, if you are accumulating area, at any moment, the change in the area is a skinny rectangle that is the height of the function itself. Thus, the derivative of the area accumulation is the function itself. I find that this idea helps them so much in solving the AP problems about this topic.