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Math 151 Section 6.2 For Area f ( x ) 0 on the interval [ a , b ] we approximate the area between the x-axis and the graph of f (x ) . We fit thin rectangles using function values for heights and non-overlapping subintervals for the bases. To describe this more clearly, we need some definitions. A partition of [ a , b ] is a set of points P is a x0 x1 x2 ... xn b . The norm of the partition = the length of the longest subinterval= max ( xi xi 1 ) i 1... n The partition is regular if the lengths of the subintervals, [ xi 1 , xi ] , are all equal and hence each subinterval length is ba . n In each subinterval [ xi 1 , xi ] choose any x-value, ci . Use f ( ci ) for the height of the ith rectangle. The ci ’s are called evaluation points. The area of the ith rectangle is f (ci )( xi xi 1 ) . When we add these rectangle areas we have n f (ci )( xi xi 1 ) is an approximation to the area under the graph of f (x ) . i 1 If f (x ) is continuous on [ a , b ] , these sums approach the true area as P approaches 0. Example 1 : Approximate the area under the graph of regular partition and 4 subintervals. a) Use c1 f ( x) x 2 over the interval [1, 3] using a 5 1, c2 , c3 2, c4 3 as evaluation points. b) 3 Use the left endpoints for evaluation points. Sketch the graph and the rectangles. c) Use the right endpoints for evaluations points. Sketch the graph and the rectangles. A better approximation to either sum in parts b and c is their average. Example where 0 1 f ( x) x f (x ) is not continuous and the sums do not have a limit. x0 x0 on [ 0,1] Use n equal subintervals and use c1 evaluation points for the other rectangles. 1 n4 . Use any appropriate