The Intuition behind The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is, as advertised, sort of the cornerstone of a first semester course in calculus, and therefore in some sense very important to understand. At first it may seem like magic, but to me it has a pretty simple intuition from considering discrete sequences.

The part of the FTC I’m going to explain is

$$ \int_a^b f’(x) \ dx = f(b)-f(a) $$

There is another part which says that \( \frac{d}{dx} \int_a^x f(t) \ dt = f(x) \). I might explain this in a later blog post if I’m ever asked to.

Now for the intuition: Consider the sequence of numbers 2, 3, 5, 8. The difference between these numbers form their own sequence, which is 1, 2, 3. This sequence of differences is kind of like the derivative at various points along a function. They represent the rate of change.

The integral is, in a sense, the aggregation of a function. To put it another way, it is kind of like summing everything together. If you take the “integral” or just the sum of the numbers 1, 2, 3, you get 6.

This is what the left-hand side of the FTC represents, by analogy. It represents the integral (sum) the derivative (the sequences of changes).

The right-hand side represents the net change, which you can think of as the difference between the end point and the starting point. In our discrete sequence this corresponds to 8-2. And wow! There it is, 6 again! Indeed these two quantities are equal.

That’s what the FTC is telling you. Integrating the changes gives you the net change.