Integration is vital to many scientific areas. Many powerful mathematical tools are based on integration. Differential equations for instance are the direct consequence of the development of integration.

So what is integration? Integration stems from two different
problems. The more immediate problem is to find the inverse
transform of the derivative. This concept is known as finding the
**antiderivative**. The other problem deals with areas and
how to find them. The bridge between these two different
problems is the Fundamental Theorem of Calculus.

What is the "**area problem**"? We want to find the area of
a given region in the plane. It is not hard to see that this
problem can be reduced to finding the area of the region
bounded above by the graph of a positive function *f* (*x*), bounded
below by the *x*-axis, bounded to the left by the vertical line
*x* = *a*, and to the right by the vertical line *x* = *b*.

The answer to this problem came through a very nice idea. Indeed,
let us split the region into small subregions which we
can approximate by rectangles or other simple geometrical figures
(whose areas we know how to compute). This is how it goes: split
the interval [*a*, *b*] into subintervals, preferably with the same
width *x*,

Let be the subregion bounded above by the graph of
*f* (*x*), bounded below by the *x*-axis, bounded to the left by
*x* = *x*_{i - 1}, and to the right by *x* = *x*_{i}, for
*i* = 1,^{ ... }, *n*.
Clearly we have

Area() = Area() + Area() + ^{ ... } + Area() .

So we focus on the subregions , for
*i* = 1,^{ ... }, *n*.
Since we want to approximate the regions by rectangles, then we
only have to worry about the upper boundary of each region (since
on the other sides we already have straight lines). Again: We are
looking for good approximations of the regions by
rectangles.

The easiest way to choose a height for our rectangles is to
choose the value of the function at the left (or right) end points
of the small intervals
[*x*_{i - 1}, *x*_{i}].

Let *L*_{i} be the rectangle defined by the left-end point and
*R*_{i} be the rectangle defined by the right-end point. Then an
approximation to
*Area*() is given by

**Example.** Consider the function

Indeed if the function *f* (*x*) is not too badly behaved, we will
show that when *n* gets larger, the numbers *LEFT*(*n*) and
*RIGHT*(*n*) get closer to
*Area*(), i.e.

Note that in the expression
*f* (*x*) d*x* the variable *x*
may be replaced by any other variable.

**Example.** Let
0. Then we have

d*x* = (*b* - *a*) .

This is true since the region is simply a rectangle.
**Example.** We have

The rectangle (depicted in red) is bounded above by *x* = *a* and its
area is *a*(*b* - *a*). The triangle (in blue) is determined by the
points: (*a*, *a*), (*a*, *b*), and (*b*, *b*). Its area is
(*b* - *a*)^{2}. So we have

A precise definition for the definite integral involves partitions and lower as well as upper sums:

**Definition.** A partition *P* of the interval [*a*, *b*] is a
sequence of numbers
{*x*_{i};*i* = 0, 1,^{ ... }, *n*} such that

For a function *f* (*x*) defined on [*a*, *b*] and a partition *P* of
[*a*, *b*], set

**Theorem.** We have

This theorem is fundamental. Let us illustrate this with the following example.

**Example.** Use the above theorem to show

(*x*_{1}^{3} - *x*_{0}^{3}) + (*x*_{2}^{3} - *x*_{1}^{3}) + ^{ ... } + (*x*_{n}^{3} - *x*_{n - 1}^{3}) = *b*^{3} - *a*^{3} .

**Exercise 1.** Use similar ideas as used in the example above
to show

d*x* ^{ . }

**Exercise 3.** Consider the function

**
**

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Helmut Knaust

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