Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

An understanding of the Fundamental Theorem of Calculus and its significance to integration. Simple examples of its use may be required in the forms:

You already know both expressions in this section, although you might not realise you know them. We will explore each expression in turn.

First:

This links the idea that integration is the reverse of differentiation and the method you are familiar with for calculating definite integrals. $F'(x) = f(x)$ tells you that when you integrate $f(x)$ you get $F(x)$; and $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$ tells you how to put limits into the expression you get once you have integrated.

A useful consequence of this expression is the following – essentially if you swap the limits around you introduce a minus sign into the integral:

You should check you can see why this works.

And the Fundamental Theorem also allows you to write integrals in other ways:

It is tempting to add the condition $a < c < b$, that is that $c$ must be between $a$ and $b$; but we have not added that condition. Can you work out why not? You do have to be a bit careful though: for instance, it might be the case that $f(x)$ is not defined at $c$ or at some other interval [i.e. part of the $x$ axis] that you might have wanted to integrate over by introducing $c$.

The second expression in the specification $\frac{d}{dx} \int_{a}^{x} f(t) \, dt = f(x)$ suggests that if you integrate and then differentiate, you should get back to what you started with [note we had to put an $x$ in the limits because if we had just put two numbers as limits, then the integral would be constant and then the differentiation would give 0]. You need to be a little careful with this expression though, as it is not as simple as it first appears. Have a look at TMUA 2020 paper 2 question 16.