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Unformatted text preview: 23. The Definite Integral P. K. Lamm 11/16/11 (09:57) p. 1 / 11 Lecture Notes: 23. The Definite Integral These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. From the material on Riemann sums in the last set of lecture notes we saw that if f is integrable on [ a,b ], then Z b a f ( x ) dx = lim n n X k =1 f ( c k ) x, where x = ( b- a ) /n , and where c k = a + k x denotes the k th right endpoint of equally-spaced intervals on [ a,b ] for k = 1 , 2 ,...,n . (Recall that we could have also let c k be left endpoints or midpoints, and that even more general Riemann sums can be used.) Terminology regarding the definite integral Z b a f ( x ) dx : We say Z is the integral sign, f ( x ) is the integrand, a is the lower limit of integration, b is the upper limit of integration, and x is the variable of integration. Example 1.1: For 0 a < b , evaluate the following definite integrals: ( i ) Z b a 1 dx, ( ii ) Z b a xdx, ( iii ) Z b a x 2 dx. (i) The definite integral in (i) represents the area under the (positive) curve y = 1, for a x b , which is just the area of the rectangle with width ( b- a ) and height 1. Thus, from the standard area formula for a single rectangle, Z b a 1 dx = b- a. (ii) The definite integral in (ii) represents the area under the curve y = f ( x ) = x for x [ a,b ], where 0 < a < b ; this curve does not go below the x-axis for these values of x . Thus the definite integral in (ii) is the difference of two areas, A 2- A 1 , where A 1 = area of the right triangle joining the points (0 , 0) , ( a, 0) , and ( a,f ( a )) = ( a,a ) , 1 23. The Definite Integral P. K. Lamm 11/16/11 (09:57) p. 2 / 11 and A 2 = area of the right triangle joining the points (0 , 0) , ( b, 0) , and ( b,f ( b )) = ( b,b ) . Using the formula for area of a triangle, A 1 = 1 2 a a = a 2 2 , A 2 = 1 2 b b = b 2 2 , so that Z b a xdx = b 2 2- a 2 2 . (iii) The definite integral in (iii) represents the area under the (nonnegative) curve y = f ( x ) = x 2 for x [ a,b ]. We cant use standard area formulas for this region, so we will need to use the definition of the definite integral to find its value. Note that f ( x ) is continuous, so we are guaranteed that the limit in this definition will exist. We will first do the special case where a = 0. Dividing [0 ,b ] into n equal subintervals and constructing rectangles with heights given by the right endpoints of these subintervals, we have x = b- n = b n , c k = 0 + k x = k b n ....
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