Definite Integrals, Objectives, Definite Integrals as a Limit of Sum

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Objectives

After studying this lesson, you will be able to:

  • Define and interpret geometrically the definite integral as a limit of sum;

  • Evaluate a given definite integral using above definition;

  • State fundamental theorem of integral calculus;

  • State and use the following properties for evaluating definite integrals :

    • if

    • if

    • if is an even function of

    • if is an odd function of .

Apply definite integrals to find the area of a bounded region.

Definite Integrals as a Limit of Sum

In this section we shall discuss the problem of finding the areas of regions whose boundary is not familiar to us. (See Fig. a)

Finding the areas of regions

Finding the Areas of Regions

  • Let us restrict our attention to finding the areas of such regions where the boundary is not familiar to us is on one side of x-axis only as in Fig. b.

  • This is because we expect that it is possible to divide any region into a few subregions of this kind, find the areas of these subregions and finally add up all these areas to get the area of the whole region. (See Fig. a)

  • Now, let f (x) be a continuous function defined on the closed interval [a, b]. For the present, assume that all the values taken by the function are non-negative, so that the graph of the function is a curve above the x-axis (See. Fig. c).

The Graph of The Shaded region

The Graph of the Shaded Region

  • Consider the region between this curve, the x-axis and the ordinates and , that is, the shaded region in Fig. c. Now the problem is to find the area of the shaded region.

  • In order to solve this problem, we consider three special cases of as rectangular region, triangular region and trapezoidal region.

  • The area of these regions base average height

  • In general for any function on

  • Area of the bounded region (shaded region in Fig. c) base average height

  • The base is the length of the domain interval. The height at any point is the value of at that point.

  • Therefore, the average height is the average of the values taken by f in . (This may not be so easy to find because the height may not vary uniformly.) Our problem is how to find the average value of in .

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