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

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.