Definite Integrals

 Differentiation deals with the rate of change while integration deals with the total change. The definite integrals are evaluated in problems relating to plane, areas, areas and volumes of solid of revolution etc.

     In this chapter, we confine ourselves to properties of definite integrals, evaluating integrals as a limit of sum and plane areas by integration.

Definite Integral as a Limit of Sum

     Let f (x) be a single valued continuous function defined in the interval [a,b] where b > 0 and let the interval [a,b] be divided into n equal parts each of length h, so that nh = b - a; then we define

Fundamental Theorem of Calculus

     The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieved.

Evaluation of definite integral by substitution

     We know that one of the most important method of evaluation of indefinite integral is method of substitution. While using method of substitution to evaluate definite integrals, following steps are involved.

     Working rule for Evaluating Definite Integral with Suitable Substitution

Some Properties of Definite Integrals

     The Properties of Definite Integrals are:

     2)

Applications of Definite Integrals

     Let y = f (x) be a curve. The area bounded by y = f (x), x-axis and the ordinates at x = a and x = b is given by

     Note 1: The area bounded by the curves f(x) and g(x) and the ordinates x = a and x = b is given by

Summary

     First Fundamental Theorem of Integral Calculus

     Let f(x) be a continuous function on the closed interval [a, b].

     Let the area function A(x) be defined by

Conclusion

     In this chapter we have studied the properties of definite integrals and application of definite integrals in evaluating the area of plane curves.