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Introduction |
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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. |
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In this chapter, we confine ourselves to properties of definite integrals, evaluating integrals as a limit of sum and plane areas by integration. |
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Definite Integral as a Limit of Sum |
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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 |
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The method of evaluating  by using the above definition is called integration from first principles. |
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Fundamental Theorem of Calculus |
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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. An important consequence, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated. |
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Let f(x) be a continuous function on the closed interval [a, b]. |
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Let the area function A(x) be defined by |
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 then
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Evaluation of definite integral by substitution |
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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. |
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Working rule for Evaluating Definite Integral with Suitable Substitution |
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Suppose we have to evaluate the integral 
(1) Let t = g(x) is the suitable substitution.
Differentiating, we get
dt = g'(x) dx
(2) Now the new variable is t.
The upper limit b and the lower limit a are in terms of x. Change these limits to the new variable g(b) and g(a).
(3) Write  and express  in terms of t.
(4) Integrate  with respect to t. |
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Some Properties of Definite Integrals |
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The Properties of Definite Integrals are: |
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Applications of Definite Integrals |
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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 |
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Note 1: The area bounded by the curves f(x) and g(x) and the ordinates x = a and x = b is given by |
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Note 2: If curve f (x) lies above the x-axis and g (x) lies below the x-axis then area bounded by f (x) and g (x) is |
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Summary |
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First Fundamental Theorem of Integral Calculus |
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Let f(x) be a continuous function on the closed interval [a, b]. |
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Let the area function A(x) be defined by |
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then
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Properties of definite integrals |
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Conclusion |
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In this chapter we have studied the properties of definite integrals and application of definite integrals in evaluating the area of plane curves. |
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We have also seen the relationship between the limit and a definite integral and learnt to evaluate limit of a sum as an integral and vice versa.
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Welcome! 
