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

Doorsteptutor material for NCO is prepared by world's top subject experts: fully solved questions with step-by-step explanation- practice your way to success.

## 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)

• 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).

• 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 .

Developed by: