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The Riemann Integral

April 5th, 2009 |

Author: Merrin

In the previous section, we saw how to approximate the area under a positive curve by adding a set of rectangles. For a finite number of rectangles, the best hope is only an approximate calculation of the area. Calculus is all about the limit and how to use it for more accurate calculations. The step we take in calculating the area in calculus is to shrink the widths of the rectangles as small as possible (one limit) while adding more rectangles to counterbalance the shrinking (another limit). In the limit as the number of rectangles goes to infinity, no matter where the tops of the rectangles intersect the curve the true area results.A simpler formula will result if we just take the upper left hand rectangles. Sometimes it is written that x* is any point in the dx interval but I find this unnecessarily complicated.

To find the area beneath f(x) between a and b we take N equal width rectangles that intersect the curve at the upper left hand corners. If the width of each rectangle is arbitrarily small in a particular limit then their width is written as dx versus capital Delta x for a finite size.

$\Delta x=\cfrac{b-a}{N} \qquad dx=\mathop{\lim }\limits_{N\to \infty } \cfrac{b-a}{N}$

The sum of the area for the upper left hand rectangle corners meeting the function is

$\displaystyle A\approx \sum _{j=0}^{N-1}f(a+j\Delta x)\Delta x$

There are two coordinated limits going on here. The width of each rectangle shrinks, but more rectangles are added. The resulting area is the Riemann integral.

$\displaystyle A=\mathop{\lim }\limits_{N\to \infty } \sum _{j=0}^{N-1}f(a+j\Delta x)\Delta x$

riemann

The Riemann sum or integral between a and b is the sum of the areas of rectangles of width dx and height f(x_i)

A much simpler notation due to Leibnitz is used for this particular calculation. The symbol for integration is an elongated S for sum. The limits of integration are indicated as a and b. The function being integrated is called the integrand.

$A=A(a,b)=\displaystyle \int _{a}^{b}f(x)dx$

Just writing A for short is ok because a and b are constants. The calculation of an area by the definite integral is a number not a function. We will show in the next section how integrals can also be functions if the limits of integration are variable.

$A(a,x)=A(x)=\displaystyle \int _{a}^{x}f(x)dx$