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Defined Integrals (continued)


The definite integral, in the examples seen, represents an area, which occurs in many cases, and is one way of presenting the definite integral.

In general, for , the area limited by f (x) and the x axis, is given by , which can represent the sum of the areas of infinite rectangles in width and whose height is the value of the function at a point in the base range:

Subdividing the range a, b into no subintervals through abscissa x0= a, x1, x2,…, Xno= b, we get the intervals (a, x1), (x1, x2),…, (Xn-1, B). At each interval (xi-1, xi) let's take an arbitrary point Hi.

Be According to the figure, the formed rectangles have an area

So the sum of the areas of all rectangles is:

which gives us an approximate value of the area considered.

Increasing the number no of subintervals such that zero is the number no from subintervals tent to infinity , we have the upper bases of the rectangles and the curve practically merging, so we have the area considered.

Symbolically, we write:

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