Unformatted text preview: The Definite Integral in this section we define the definite integral and learn some ways to evaluate it using the summation formulas below and using areas of geometric figures. Summation Formulas 1. 2. 3. 4. Example: Example: lim lim lim lim Example: lim lim Determine the area of the region bounded by the curve in MTH 173 section 5.2 notes page 1 and above the -axis. area to determine left point rectangles Using rectangles, each has a width of , since the total width of the interval on the axis is .
-1=x 0 x 1 x2 x3 xi-1 x i xn-1 xn =1 from to there are subintervals Using left rectangles, we have and, in general, replacing the specific number subscript with the generic as a counter, we have, Also, then, replacing with , we have The left endpoint of rectangle number 1 is given by , the left endpoint of
MTH 173 section 5.2 notes page 2 rectangle 2 is given by , the left endpoint of rectangle 3 is given by general, the left endpoint of rectangle number is given by The area of rectangle number is given by its height , and in times width , so lim lim Expand in the parens, rite result in descending powers of : lim lim lim When we approximate a definite integral with a few rectangles (as in section 1), the most accurate approximation is usually obtained by using rectangles with the height of each determined by evaluating the function at thw midpoint of each subinterval, as below: Example: , midpoint rule with MTH 173 section 5.2 notes page 3 From the graph, we see that the coordinates of the midpoints are and , and the width of each interval is , so the area of these 4 rectangles is given by Definition of a Riemann Sum Let be defined on the closed interval given by b and let be a partition of where is the length of the th subinterval. If subinterval, then the sum is any point in the th is called a Riemann sum of for the partition . is the norm of the partition Defn. The length of the largest subinterval of a partition and is denoted by Defn of a Definite Integral If is defined on a closed interval lim exists, then lim is integrable on and the limit and the limit is denoted by This limit is called the definite integral of from to . The number is the lower limit of integration, and the number is called the upper limit of integration. The Definite Integral as the Area of a region MTH 173 section 5.2 notes page 4 If is continuous and nonnegative on the closed interval , then the area of the region bounded by the graph of , the axis, and the vertical lines and is given by Area 36. Since the region is a semicircle of radius . The area of a circle of radius 2 is so the area of the semicircle is , so Evaluate using geometry, areas 34. Given a. directly from area: ! MTH 173 section 5.2 notes page 5 b. from the negative of the area, as the function is always negative in this interval, ! c. here we add separate the integral into three intervals, calculate the value of each, and add the integrals' values: ! MTH 173 section 5.2 notes page 6 ...
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This note was uploaded on 09/17/2009 for the course MTH Calc taught by Professor Bush during the Spring '09 term at Northern Virginia.
- Spring '09