7.2.2 The First Half of the Fundamental Theorem:
         Evaluation of Definite Integrals

Our next task is to show that the relationship described in the equation

is not special to velocity-distance calculations but holds for integrals in general. To do this, we consider an approximate solution to the initial value problem

by means of Euler's Method.

Recall that Euler's Method generates a sequence of numbers `s_1`, `s_2`, `s_3`, and so on, that approximates values of `stext[(]t text[)]` at times `t_1`, `t_2`, `t_3`, and so on. Specifically, we can divide the interval `[a,b]` into `n` subintervals by taking `Delta t=text[(]b-atext[)/]n` and choosing the points of subdivision to be

	`t_0`	`=a`,
	`t_1`	`=a+Delta t`,
	`t_2`	`=a+2 Delta t`,
	`t_3`	`=a+3 Delta t`,
and so on, up to
	`t_n`	`=b`.

Then, for `k=1, 2, ... , n`, [starting with `s_0=stext[(]0 text[)]=0`], each `s_k` is generated from the previous one by the "rise `=` slope `times` run" formula:

To make the connection between the Euler's Method approximate solution to the initial value problem and the definite integral in

we write out explicitly the values of the `s_k`'s:

	`s_1=s_0+vtext[(]t_0 text[)]Delta t	`=vtext[(]t_0 text[)]Delta t`,
	`s_2=s_1+vtext[(]t_1 text[)]Delta t`	`=vtext[(]t_0 text[)]Delta t+vtext[(]t_1 text[)]Delta t`,
	`s_3=s_2+vtext[(]t_2 text[)]Delta t`	`=vtext[(]t_0 text[)]Delta t+vtext[(]t_1 text[)]Delta t+vtext[(]t_2 text[)]Delta t`,
and so on, up to
	`s_n=s_(n-1)+vtext[(]t_(n-1) text[)]Delta t`	`=vtext[(]t_0 text[)]Delta t+vtext[(]t_1 text[)]Delta t+ cdots +vtext[(]t_(n-1) text[)]Delta t`.

In sigma notation, this last equation is the same as

That is, `s_n`, the Euler approximation to `stext[(]t_n text[)]` [which is the same as `stext[(]b text[)]`], is exactly equal to the left-hand sum with `n` terms that approximates the definite integral of `vtext[(]t text[)]` from `a` to `b`.

Now, as `n` becomes large — and the step size `Delta t` becomes small — two things happen to the equation

First, the Euler approximation to `stext[(]t text[)]` gets closer and closer to the true solution of the initial value problem

In particular, `s_n` approaches `stext[(]b text[)]`. Second, the sum on the right-hand side approaches the definite integral of `vtext[(]t text[)]` from `a` to `b`. When we consider limiting values, we find

This observation is not limited to the relationship between distance and velocity. In particular, nowhere in this discussion did we use any particular physical interpretation of the functions `vtext[(]t text[)]` and `stext[(]t text[)]`.

Checkpoint 1

We are now ready to examine the relationship between antidifferentiation and the definite integral. Suppose we want to calculate ${\int }_a^{b}f\left(t\right)dt$ and we know one antiderivative for `ftext[(]t text[)]` is `Gtext[(]t text[)]`. Then the solution of the initial value problem

is a function that differs from `Gtext[(]t text[)]` by a constant. That constant might be zero, but all we know for sure is that `Ftext[(]t text[)]=Gtext[(]t text[)]+C` for some constant `C`. Since `Ftext[(]atext[)]=0`, we know that `0=Gtext[(]atext[)]+C`, or `C=-Gtext[(]atext[)]`.

Checkpoint 2

The last equation in Checkpoint 2,

is part of the Fundamental Theorem of Calculus. It says that one way to find the value of a definite integral ${\int }_a^{b}f\left(t\right)dt$ is to find any antiderivative `Gtext[(]t text[)]` of `ftext[(]t text[)]` and then calculate `Gtext[(]b text[)]-Gtext[(]atext[)]`.

Example 1

Calculate ${\displaystyle {\int }_0^1{t}^{3}dt}$.

Solution   We need a function that has `t^3` as its derivative. You already know that one such antiderivative is `t^4text[/]4`. It follows that

Note that we could have used `t^4text[/]4+7` as an antiderivative, in which case the computation would have been

Of course, if we had used any other antiderivative of `t^3`, the result would have been the same.