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Definite Integral

Definite Integral

A definite integral is the area beneath a function. This area can be reckoned by summation a very small rectangles. It is an approximation of the area of the outer rectangles and inner rectangles. If there is no difference of both rectangles areas left, then you get the area of the wanted integral.

The rectangle between these two rectangles can be used to evalutate the whole area of a function.

Approximations to integral of  from 0 to 1, with ■ 5 right samples (above) and ■ 12 left samples (below) (Source: KSmrq)

Sum of many rectangles

The area of an rectangle is length multiplied by width. In this formula  represents the height (length) of the rectangles which shall be summed up. The variable  is the constant width of the rectangles.

The Greek letter  is an abbreviation of the word “difference”. The width of the rectangle x is always constant.

The length (= height in a function) of the rectangles can change, but their width always has the value , a constant small number.

The last index n shows, there are a plenty of rectangles which are to be summed up. Such a letter number is used to express great numbers.

 is a notation, you must understand.

The index i shows there are many different x. The names of this different x are , ,  etc. and these x-variables represent certain numbers.

For instance:

etc.

So, put these different values into the above fomular. The width of the rectangle  shall be 1.

Formular of an rectangle in a function

Fill in the number 1 for the width  :

Now replace  by numbers:

This was an unusual way to determine the area of an rectangle. If you think this procedure to be continued to infinitiy then the -formular has be complimented.

Here the index n goes to infinity.

Gauss invented a new sign for this formula, a long S. This is the abbreviation of sum.

Carl Friedrich Gauss (Source: Gottlieb Biermann, A. Wittmann)

The long S is an elegant writing for the integral  (“whole reckoning”) . It demonstrates what is behind the integral, a pure summation of many rectangles.

Instead of the difference  the integral only knows the differential dx, which is very small, so small that you can’t imagine.

The index has become the boundaries of the function. The a is the lower boundary and the b is the upper boundary of the x-axis.

The variable behind the differential d can almost be every letter of the alphabet. It indicates, which variable of the function shall be integrated.

For instance:

A nice formula in physics, the definition of energy per pulse.

Here the integral variable is t. Even if you don’t know the formula, you can interpret it. The boundaries are 0 and T. The area of the function lies between these boundaries and the infinite small rectangles are  which shall be summed up by .