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The Definite Integral Concepts: 1. A partition of the interval [a,b] is a finite subset of the interval that contains the endpoints (i.e., a partition P = {a = x0 < x1 < x2 < . . . < xn = b} where n is a counting number and xi belongs to [a,b] for each i, 0 i n). A partition is a regular partition if and only if the distance between any two consecutive points in the partition is always the same, or all of he subintervals determined by the points in the partition are the same length. . In other words, the partition P = {a = x0 < x1 < x2 < . . . < xn = b} is regular if and only if xk - xk-1 = (b - a)/n for each k, 1 k n. Note that xk is a symbolic way of representing xk - xk-1 . The mesh or norm of a partition is the length of its largest subinterval. The norm of a partition P is denoted by P. In other words, if P = {a = x0 < x1 < x2 < . . . < xn = b}, P = max { xk | k = 1, 2, . . . , n}. Examples: a. P = {0, 1, 2, 4, 8} is a partition of the interval [0,8], but it is not regular. Also, note that P = 4. b. P = {0, 3, 6, 8} is a partition of [0,8] that is not regular. P = 3. c. P = {0, 2, 4, 6, 8} is a regular partition of [0,8] with P = 2. 2. Let f be a function defined on the interval [a,b]. Let P = {a=x0 < x1 < x2 < . . . < xn=b} be a partition of [a,b]. Select a number in each of the subintervals determined by P; i.e., choose uk [xk-1,xk] for each k, 1 k n. The sum, n f (u k=1 k )( x k - x k -1 ) , is called a Riemann sum for the function f determined by the partition P. z b 3. The definite integral of a function f from x = a to x = b , denoted f(x) dx , is a the limit of the Riemann sums for f as the norm or mesh of the determining partitions approaches 0, provided the limit exists. If the limit exists, f is called integrable on the interval. If P is a regular partition of [a,b] that divides the interval into n subintervals, then z b n f(x) dx = lim f( uk )( x k - x k -1 ) where uk [xk-1,xk] for each k, 1 k n. n a k=1 The Definite Integral Page 2 4. Geometrically, the definite integral of a function f from x = a to x = b where a < b, z b a f ( x )dx , represents the area of the region bounded by the graphs of y= f(x), the x-axis, and the lines x = a and x = b that lies above the x-axis minus the area bounded by the same graphs that lies below the x-axis. Example: z 2 1 3 8 ( x + 3)( x - 2)( x - 5)dx = - 1375 96 (or approximately -14.32292) is the negative of the area of the shaded region below the x-axis in the figure to the right. z 1 2 8 5 ( x + 3)( x - 2)( x - 5)dx = 117 32 (or approximately 3.65625) is the area of the shaded region above the x-axis in the figure to the right. z 5 1 32 ( x + 3)( x 2)( x 5)dx = - is the -3 8 3 area of the shaded region above the x-axis minus the area of the shaded region below the x-axis (or the sum of the two numbers above). 5. If an object is moving along a line with velocity at time t given by v(t), then the total z b distance traveled by the object from t = a to t = b is given by |v(t)| dt . a The Definite Integral

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posted: | 3/7/2012 |

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