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Unformatted text preview: 5.1: Areas and Distances 5.2: The Definite Integral (Dated: September 21, 2011) THE AREA PROBLEM Part of the area problem is to make the intuitive idea of what the area of a region is precise by giving it an exact def- inition of area. This is easy for a regions with straight sides such as a rectangle, a triangle, and a polygon. However, it isnt easy for a region with curved sides. Under appropri- ate stipulations, the area of a region with curved sides can be defined as the limit of the sum of the areas of its non- overlapping constituents which are regions of straight sides. By the region S that lies under a continuous curve y = f ( x ) from a to b is meant the region bounded by the graph of a continuous function f , where f ( x ) 0, the vertical lines x = a and x = b , and the x-axis. Lets subdivide S into n strips S 1 , S 2 , , S n of equal width x = b- a n . These strips divide the interval [ a , b ] into n subintervals [ x , x 1 ], [ x 1 , x 2 ], [ x 2 , x 3 ], , [ x n- 1 , x n ], where x = a and x n = b . Exercise 1 Verify that x i = a + i x for each i, i = 0,1,2 , n. Let x * i be any number in the i th subinterval [ x i- 1 , x i ], called a sample point. One can then approximate the i th strip S i by a rectangle of width x i- x i- 1 = x and height f ( x * i ). It follows that n X i = 1 f ( x * i ) x = f ( x * 1 ) x + f ( x * 2 ) x ++ f ( x * n ) x (1) is an approximation of the area of S . Such a sum is called a Riemann sum. If x * i = x i- 1 , the left endpoint of the i th subinterval [ x i- 1 , x i ], then the Riemann sum is called a left sum, denoted L n ; If x * i = x i , the right endpoint of the i th subinterval [ x i- 1 , x i ], then the Riemann sum is called a right sum, denoted R n ; if x * i = x i = x i- 1 + x i 2 , the midpoint of the i th subinterval [ x i- 1 , x i ], then the Riemann sum is called a middle sum, denoted M n . The rule of evaluating the middle sum is called the midpoint rule. Since f is continuous, the limit A ( S ) = lim n n X i = 1 f ( x * i ) x (2) always exists and is defined as the area of the region S ....
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