The Fundamental Theorem of Calculus

What an amazing breakthrough this theorem provided!

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 of 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.'

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.

It should become much clearer in the video (where I can talk, wave my hands around, and draw pictures)!

The next challenge is how to actually find primitives for a range of functions. Subsequent videos will address that issue.

I hope you enjoy the video.

Best wishes,

Graeme

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Thank you.