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The Definite Integral
The integral is defined by the area under the curve of a function's graph. It is most commonly found by the concept of many rectangles under the line and as their number increases, the area of the rectangles becomes closer to the area under the curve.

Let $$f$$ be a function defined on the closed interval $$[a, b]$$. Divide this interval into n subintervals by choosing any $$n-1$$ intermediate points between $$a$$ and $$b$$. Let $$x_{0} = a$$ and $$x_{n} = b$$, and let $$x_{1}, x_{2}, ..., x_{n-1}$$ be the intermediate points so that

$$x_{0} < x_{1} < x_{2}< ...< x_{n-1}< x_{n}$$

The points $$x_{0} < x_{1} < x_{2}< ...< x_{n-1}< x_{n}$$ are not necessarily equidistant. Let $$\Delta_{1}x$$ be the length of the first subinterval so that $$\Delta_{1}x = x_{1}-x_{0}$$; let $$\Delta_{2}x$$ be the length of the second subinterval so that $$\Delta_{2}x = x_{2} - x_{1}$$; and so forth so that the length of the $$i$$th subinterval $$\Delta_{i}x$$, and

$$\Delta_{i}x = x_{i} - x_{i-1}$$

A set of all such intervals of the interval $$[a, b]$$ is called a partition of the interval $$[a, b]$$. Let $$\Delta$$ be such a partition. The partition $$\Delta$$ contains $$n$$ subintervals. One of these subintervals is the longest; however, there may be more than one such subinterval. The length of the longest subinterval of the partition $$\Delta$$, called the norm of the partition, is denoted be $$\vert \vert \Delta \vert \vert$$.

Choose a point in each subinterval of the partition $$\Delta$$: Let $$\xi_{1}$$ be the point chosen in $$[x_{0}, x_{1}]$$ so that $$x_{0} \leq \xi_{1} \leq x_{1}$$. Let $$\xi_{2}$$ be the point chosen in $$[x_{1}, x_{2}]$$, so that $$x_{1} \leq \xi_{2} \leq x_{2}$$, and so forth, so that $$\xi_{i}$$ is the point chosen in $$[x_{i-1},x_{1}]$$, and $$x_{i-1} \leq \xi_{i} \leq x_{i}$$. Form the sum

$$f(\xi_{1} )\Delta_{1}x + f(\xi_{2} )\Delta_{2}x + ... + f(\xi_{i} )\Delta_{i}x + ... + f(\xi_{n} )\Delta_{n}x$$

or

$$ \sum^n_{i=1}f(\xi_{i})\Delta_{i}x $$

Such a sum is called a Riemann sum, named for the mathematician Georg Friedrich Bernhard Riemann (1826-1866).

Taking the norm to be sufficiently small, the sum can be made accurate to the area under the curve. To find the exact area of the region, let the region be denoted as $$R$$, and the norm must approach zero. Using the limit and Riemann sum, we have

$$\lim_{ \vert \vert \Delta \vert \vert \rightarrow 0} f(\xi_{i})\Delta_{i}x$$

This is the definition of the definite integral, one of the most important aspects in calculus, and is written as

$$\int_{a}^{b}f(x)dx=\lim_{ \vert \vert \Delta \vert \vert \rightarrow 0}\sum^n_{i=1}f(\xi_{i})\Delta_{i}x$$