Starting Integral Calculus by the Summation of Riemann Integrals

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Rigourous Riemann Integrals

To attempt any rigour in progressing from these approximations, we need to use Bernhard Riemann’s work on “Riemann Integrals”.

To Define A Finite Partition of Real Numbers

Riemann’s work starts with a “finite partition” in the set of real numbers. Let’s set “x(0) = a < x(1)…< x(n) = b”. Then we can create a partition of [a, b] as a set of ‘n’ sub-intervals [x(0), x(1)], [x(1), x(2)],… [x(n-1), x(n)]. The “n+1” points form a partition, ‘P’, and we can say P = {x(0), x(1),…x(n)}.

Each subinterval [x(i), x(i+1)] has the length L = x(i+1) – x(i). The largest length of a subinterval of ‘P’ is called the “norm of P”, and is shown as “||P|| = maximum for all ‘i’ of ( x(i+1) – x(i) )”.

To Define a Refinement of a Partition

We will need to “refine a partition”, or slice it thinner. Say that P’ is a partition over the same [a, b] as partition P, with more points x'(i). If every point x(i) in the original partition P is also a point in P’, but there are additional points in P’, then P’ is a refinement of P.

To Define a Riemann Sum of a Partition

In case the web site does not support the “less than or equal” sign, it is shown as “<=”.

Let f(x) be a function, ‘f’, defined for all points in a partition P. Let {c(i)} be a set of real numbers in the partition P, such that x(i) <= c(i) <= x(i+1).

The Riemann sum, σ (“sigma”) of this function is the sum from j = 0 through (n-1) of f( c(j) )*( x(j+1) – x(j) ).

“Image of Riemann Sum Formula” Image by Mike DeHaan

It is important to realize that the c(j) values may take any value from x(j) to x(j+1), so there is an infinite number of values for a Riemann sum.

Return of the “Delta-Epsilon Limit”

As in the previous article on differential calculus, we will use “delta” (‘δ’) to mean a small change in ‘x’, and “epsilon” (‘ε’) to mean the change in the function f(x) over that δ range.

Let f(x) be a function defined on the real numbers in partition P of [a, b]. Let σ be any Rieman sum of function ‘f’ on P. Let δ and ε be small real numbers, greater than zero. Let L be a real number, such that the absolute value |σ – L| < ε when the maximum subinterval ||P|| < δ.

When all these conditions are met, ‘L’ is the Riemann integral of the function ‘f’ over the partition ‘P’ from [a, b]. The symbol for “integral” follows.

"Image of Parabola Showing Exact Area Under the Curve" by Mike DeHaan

“Image of Parabola Showing Exact Area Under the Curve” by Mike DeHaan

The Simplest Example of a Riemann Integral

If “f(x) = a” where ‘a’ is a constant real number, then any Riemann sum σ of “f = sum of( a*( x(j+1) – x(j) ) )”. That is simply “a*( ( x(n) – x(n-1) ) + ( x(n-1) – x(n-2) ) +…+ ( x(2) – x(1) ) + ( x(1) – x(0) ) )”.

But the right side has many cancelling pairs of “… – x(n-1) + x(n-1)…”, so it leaves “a*( ( x(n) – x(0) ) )”.

Amazingly, this is exactly what we found in the first examples where the rate was a constant value, and the start (now shown as “x(0)”) was zero.

This article needs to close before examining more complicated integration.

Closing Notes for the Riemann Integral

Mathematicians would add that this Riemann integral from ‘a’ to ‘b’ exists if and only if the function ‘f’ is “Riemann integrable” on the partition [a, b].

That should serve as a warning: some functions cannot be integrated. Remember that it is impossible to find the differential of some functions, too. A typical example would be “y = 1/x”, where (0,?) is undefined.

Who was Bernhard Riemann?

Bernhard Riemann: Image by Ævar Arnfjörð Bjarmason

Bernhard Riemann was born in Hannover in 1826. After graduating from Göttingen University, he began studying theology but transferred to philosophy in order to pursue his long-evident mathematical gifts.

After brief studies elsewhere, Riemann returned to Göttingen to complete his career. He addressed a number of important issues, including the introduction of an “n-dimensional Riemannian manifold” that Albert Einstein would use, sixty years in the future, in his theory of general relativity.

His greatest work might be the still-unproven conjecture called the Riemann hypothesis. It has significant implications for the distribution of prime numbers.

Riemann married in July of 1862. Later that year, a common cold developed into tuberculosis. He died in Italy in 1866, survived by his wife and a daughter, aged three.

“Integral Symbol” Image by M. DeHaan

References:
Hoffman,Mike. Bernhard Riemann. US Naval Academy. (Sept. 29, 2009). Accessed Aug. 6, 2011.
Trench, William F. Introduction to Real Analysis. Trinity University, San Antonio. (2003). PDF accessed Aug. 6, 2011.

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