Question

How do you use the fundamental theorem of calculus and what can it be used for?

Answer 1

The main application is to find exact answers for various kinds of definite integrals. Another application is that it allows us to construct antiderivatives (in a theoretical way).

Explanation:

Here's a simple example where the Fundamental Theorem of Calculus allows us to find the value of a definite integral. Let `f(x)=x^3`. Since `F(x)=x^4/4` is an antiderivative of `f(x)` (meaning `F'(x)=f(x)` for all `x`), we can say that, for example,

`int_{-1}^{3}x^3\ dx=int_{-1}^{3}f(x)\ dx=F(3)-F(-1)`

`=3^4/4-(-1)^4/4=81/4-1/4=80/4=20`

An example of how the Fundamental Theorem of Calculus can be used to construct an antiderivative. Let `f(x)=cos(x^2)`. Is there a function `F(x)` so that `F'(x)=f(x)` for all `x`? There is, but there is no way to represent the function `F(x)` in terms of so-called "elementary functions ".

However, if we let the upper limit of integration of `f` be a variable, this defines, in a theoretical way, an antiderivative of `f`. In particular, `F(x)=int_{0}^{x}cos(t^2)\ dt` is an antiderivative of `f` (the lower limit of zero is arbitrary. Any other number gives another antiderivative of `f`.) The Fundamental Theorem of Calculus guarantees this (that `F'(x)=f(x)=cos(x^2)` for all `x`).

What good is this function? Evidently it is important enough in applications to be given a name. It's a "Fresnel function " and it has applications to optics and highway design.

Can it be approximated and graphed? Yes. Use numerical integration approximation techniques like Simpson's Rule to find values of `F(x)`. If you do this for many values of `x`, you can graph this function. A graph of `F(x)` is shown below.

As an exercise, see if you can find the critical points and approximate local extreme values of this function.