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UNIT4 - Unit 4 The Denite Integral The formal denition of...

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Unit 4 The Definite Integral The formal definition of the definite integral Suppose the function f is continuous on the interval [a,b]. We begin by partitioning the interval [a,b] into n subintervals of equal length by choosing points, say x 0 , x 1 , ...x n between a and b subject to the conditions that a = x 0 < x 1 < x 2 < ... < x n 1 < x n = b (1) and x = x i x i 1 = ( b a ) /n (2) We then construct a rectangle over each interval [ x i 1 , x i ] of “height” f ( x i ). The area of the i’th rectangle is therefore the positive or negative value (de- pending if f ( x i ) is positive or negative) of f ( x i ) × x . See figure 1: As the partition of [a,b] becomes finer and finer ( i.e. as n gets bigger and so our rectangle widths become smaller) we would expect the rectangles to approximate the region between the x-axis and the curve with increasing accuracy. This is indeed what happens and we define the definite integral of f(x) between a and b in this manner: Definition 4.1. Let f(x) be a continuous function on the interval [a,b], then with the above notation the definite integral of f(x) between a and b is then defined as: b a f ( x ) dx = lim n →∞ [ f ( x 1 )∆ x + f ( x 2 )∆ x + ...f ( x n )∆ x ] = lim n →∞ n i =1 f ( x i )∆ x (3) 1
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y = f(x) y x = b x x i-1 i f (x ) i a=x x x x x x x x n-2 n-1 n 0 1 2 i-1 i Figure 1: Of course, calculating such a limit may be quite difficult, luckily there is an easy way of calculating the definite integral of a function if we know an antiderivative of the function.The following theorem is called the fundamen- tal theorem of calculus,it gives an easy way of calculating definite integrals and should be studied carefully. its ramifications shall be felt throughout the remainder of the course. Theorem 4.2. The Fundamental Theorem of Calculus Suppose f ( x ) is continuous on the interval [a,b], and that F ( x ) is an an- tiderivative of f ( x ) on [a,b]. Then: b a f ( x ) dx = F ( b ) F ( a ) 2
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or equivalently b a F ( x ) dx = F ( b ) F ( a ) Definition 4.3. The numbers a, and b above are called the limits of inte- gration .
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