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We will develop the definite integral as a means to calculate the area of certain
regions in the plane. Given two real numbers *a* < *b* and a function *f* (*x*) defined
on the interval [*a*, *b*], define the region *R*(*f*, *a*, *b*) to be the set of points
(*x*, *y*) in the plane with *a*≤*x*≤*b* and with *y* between 0 and *f* (*x*).
Note that this region may lie above the *x*-axis, or below, or both, depending on
whether *f* (*x*) is positive or negative. In computing the area of *R*(*f*, *a*, *b*), it
will be convenient to count the regions above the *x*-axis as having "positive
area", and those below as having "negative area".

Figure %: Computing the Area of *R*(*f*, *a*, *b*)

We can split up the interval [*a*, *b*] into *n* smaller intervals (for some integer
*n*) of width *Δx* = (*b* - *a*)/*n*. Let

s_{i} = a + i(Δx) |

for *i* = 0, 1,…, *n*, so that the *n* intervals are given by [*s*_{0}, *s*_{1}],…,[*s*_{n-1}, *s*_{n}].

Figure %: Splitting up the interval [*a*, *b*] into *n* smaller intervals

Let *M*_{i} be the maximum value of *f* (*x*) on the interval [*s*_{i-1}, *s*_{i}].
Similarly, let *m*_{i} be the minimum value of *f* (*x*) on the interval
[*s*_{i-1}, *s*_{i}]. Consider the region made up of *n* rectangles, where the *i*-th rectangle is
bounded horizontally by *s*_{i-1} and *s*_{i} and vertically by 0 and *M*_{i}. As
shown below, this region contains *R*(*f*, *a*, *b*).

Figure %: Containing *R*(*f*, *a*, *b*) with rectangles

Moreover, we know how to compute the area of this region. It is simply

(M_{1}) + (M_{2}) + ^{ ... } + (M_{n}) = M_{i} |

We denote this *n*th upper Riemann sum by *U*_{n}(*f*, *a*, *b*). Replacing *M*_{i} in the above
with *m*_{i}, we obtain a region contained in *R*(*f*, *a*, *b*).

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