If you are currently studying calculus at school or, if you never understood it but would like to see if it makes sense now, this video is designed for you. It is not designed for the complete novice. I assume some understanding of graphing and differentiation. At a later time I intend to explain the foundations of calculus 'from the ground up.'
For many years (at least since Archimedes, around 250 BC), mathematicians had known that areas of unusual shapes could be calculated by imagining that they were divided into innumerable infinitesimally thin strips and adding all those 'parts' together. They had also known how to calculate gradients and rates (comparing two quantities by dividing them) and, with the development of calculus, how to calculate them anywhere on a curve by differentiating the function.
The Fundamental Theorem of Calculus showed that these two concepts were intimately related! In fact, finding the area under a curve proved to be the exact opposite to finding its gradient! That is, if the process for finding a gradient is called differentiating, the process for finding the area under a curve is called antidifferentiating.
Suddenly, this made finding areas much easier! Antidifferentiating is also called finding primitives or integrating.
This understanding is the result of the first part of the Fundamental Theorem of Calculus. The second part of the theorem shows that the exact area under a curve and between two boundary values on the x-axis is found by substituting those boundary values into the primitive and finding the difference between the two results.