In this section we explore the connection between the Riemann and Newton integrals. Note that these two integrals are very different in nature. To start with, the Riemann integral is a definite integral, therefore it yields a number, whereas the Newton integral yields a set of functions (antiderivatives). The Riemann integral is a geometric notion (area), while the Newton integral is an algebraic notion. Finally, these integrals apply to different sets of functions. As we will see, both can be applied to "nice" functions; on the other hand, for instance the jump function we saw before is Riemann integrable, but it does not have an antiderivative.

It may therefore come as a surprise that in fact, there is a deep connection between these two. This is the topic of this section. As one of the consequences we will find a convenient way of evaluating definite integrals.

We start with a definition. Let *f* be a function that is Riemann
integrable on an interval *a*,*b*].*c* from *a*,*b*].*f* is also
Riemann integrable on *c*,*x*]*x* from
*c*,*b*]*x*,*c*]*x* from *a*,*c*].*x* from *a*,*b*]

Note that since we used *x* as an upper limit, we cannot use it as a
variable in the integral and had to choose another letter. The value of
*F*(*x*)

Since we can define this number *F*(*x*) for all *x* from
[*a*,*b*], we obtained a function on [*a*,*b*] in this
way. For a possible interpretation of this integral, click
here. We have the following:

Theorem(The Fundamental Theorem of Calculus I, TFC 1).

Letfbe a function that is Riemann integrable on[ leta,b],cbelong to[ Fora,b].xfrom[ definea,b],Then

Fis a continuous function on[ Moreover, fora,b].xfrom( ifa,b),fis continuous atx, thenFis differentiable atxandF′(x) =f(x).

**Example:** Consider the function
*f* (*x*) = *x* + 1

Let *c*=1. If *x* ≥ 1,

If *x* ≤ 1,

Thus on

Indeed, we now see that *F* is continuous on *F* ′(*x*) = *x* + 1 = *f* (*x*),

Since most of the time we deal with continuous functions and these are Riemann integrable, we have this weaker but useful version:

Theorem(The Fundamental Theorem of Calculus I, TFC 1).

Letfbe a continuous function on[ leta,b],cbelong to[ Fora,b].xfrom[ we definea,b]Then

Fis an antiderivative offon[ a,b].

We now have the first connection. We see that a continuous function is both Riemann integrable and Newton integrable, and we can get an antiderivative (the Newton integral) using the Riemann integral. There is a connection the other way, too:

Theorem(The Fundamental Theorem of Calculus II, TFC 2).

Letfbe a continuous function on[ Ifa,b].Fis an antiderivative offon[ thena,b],

This is also called the **Newton-Leibniz Formula**. Since finding an
antiderivative is usually easier than working with partitions, this will be
our preferred way of evaluating Riemann integrals. Since this is used so
often, in calculations we will be also using this convenient notation:

**Example:** We know that
*F*(*x*) = 3*x*^{2} + *x* − 3*f* (*x*) = 6*x* + 1*F* ′ = *f* )*f* is continuous.
Therefore we can evaluate

Recall that when we find a Newton integral, we express it as
*F*(*x*) + *C*.*C* cancels. This is the
reason why we simply ignore constants in antiderivatives when evaluating
definite integrals. It is easier, for instance in the above calculation we
would prefer to write

The Newton-Leibniz formula can be also used in case when
*a* > *b*.

The two Fundamental Theorems show that for a continuous function, the Riemann and Newton integrals are somehow connected. A more general statement is also true.

That strange integral at the beginning did not just come out of the blue, in fact it has a natural interpretation in physics. The Fundamental theorem then follows from elementary physical reasoning, it is a very useful take on this topic and we offer it in this note. Here we will show a useful mathematical interpretation.

Recall that given a function, if we first integrate it and then
diffferentiate, we arrive at the same function as in the beginning.
However, if we first differentiate and then integrate, then this no longer
works (see
Newton integral in
Integrals - Theory - Introduction). However, when we look at our results
here not from the point of view of *f*, but we focus on *F*, we
get a very interesting formula:

So if we differentiate a function, we can recover it by integrating it if we use the definite integral in the right way.