Understanding Riemann Integration in Mathematics
Exploring Riemann Integration in mathematics involves concepts like partitions of intervals, upper and lower Riemann sums, graphical representations, and refinements of partitions. This study guide delves into the definitions, calculations, and applications of Riemann Integration, providing a comprehensive overview for S.Y.B.Sc students.
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Rayat Shikshan Rayat Shikshan sanstha s sanstha s Arts Science and Commerce College, Mokhada Arts Science and Commerce College, Mokhada District Palghar 401 604 District Palghar 401 604 Topic Name : Riemann Integration Topic Name : Riemann Integration Class : S. Y. B. Sc. Semester : III Class : S. Y. B. Sc. Semester : III Prepared by : Mr. Prepared by : Mr. Patil Patil Prashant Prashant K. K.
Riemann Integration Graphical Representation Graphical Representation
Partition of Interval Let I = [a, b] be an interval. Let a = x0 < x1< < xn = b be points of the interval [a, b]. The set P = {a = x0, x1, , xn = b}is called Partition of the interval [a, b] and it divides the interval into n subintervals. [x0, x1], [x1, x2], .. , [xn-1, xn]. In general we denote subintervals as [xr-1, xr], r = 1, 2, , n. x2 xr -1 xr a x1 b Partition of [a, b]
Definition of Upper and Lower Riemann Sum Let f(x) be the bounded function in the given interval [a, b] let P = {a = x0, x1, , xn = b} be the partition of [a, b]. Let Ir = [xr-1, xr] be subintervals and let mr & Mr be the Infimum and Supremum of f(x) in each subintervals respectively. Let xr= xr xr-1 be the length of each subintervals Ir. Then Lower Riemann Sum L(P, f) = mr xr Upper Riemann Sum U(P, f) = Mr xr
Upper Riemann Sum Lower Riemann Sum
Refinement of the Partition Let I = [a, b] be an interval and let P & Q be two partitions of [a, b]. Then Q is called refinement of partition P or Q is finer than P if P Q. This means that Q has more number of points than that of P. Example: Let I = [0, 1] Let P = {0, 0.5, 1} & Q = {0, 0.25, 0.5, 0.75, 1} Here we can observe that partition Q has more number of points than P. So Partition Q is called refinement of partition P
Upper Riemann Sum Lower Riemann Sum Here n = 5
Upper Riemann Sum Lower Riemann Sum Here n = 10
Upper Riemann Sum Lower Riemann Sum Here n = 50
Upper & Lower Riemann sums-Example with n = 5, 10, 50 subintervals of equal length
Theorem Statement : Let I = [a, b] be an interval. Let f be a bounded function defined on I. Let P be any partition of [a, b]. Let mr = inf x [xr 1,x?]f(x) and Mr = m(b a) L(P, f) U(P, f) M(b a) Proof : Let P = {a = x0, x1, , xn = b} be any partition of [a, b] dividing [a, b] into n subintervals Ir = [xr-1, xr] for r = 1, 2, ., n Let mr = inf x [xr 1,x?]f(x), then sup x [xr 1,x?]f(x) and Mr = x [xr 1,x?]f(x) for r = 1, 2, sup ., n
Clearly, m mr Mr M Where, m = inf x [?,?]f(x) and M = sup x [?,?]f(x) m(xr - xr-1) mr(xr - xr-1) Mr(xr - xr-1) M(xr - xr-1) n n n m(xr xr 1) r=1 M(xr xr 1) mr(xr xr 1) r=1 Mr(xr xr 1) r=1 r=1 n n n n (xr xr 1) r=1 (xr xr 1) mr(xr xr 1) r=1 Mr(xr xr 1) m r=1 M r=1 n n n m(xn x0) r=1 mr(xr xr 1) r=1 Mr(xr xr 1) M(xn x0) m(b a) L(P, f) U(P, f) M(b a) Hence the proof.
Upper and Lower Riemann Integrals By previous theorem we can find that the set{U(P, f)/P is a partition of [a, b]}is certainly bounded below by m(b a) and has greatest lower bound. Also we can observe that the set{L(P, f)/P is a partition of [a, b]}is certainly bounded above by M(b a) and has least upper bound bf(x) dx Let U(f) = glb{U(P, f)/P is a partition of [a, b]} = a And L(f) = lub{L(P, f)/P is a partition of [a, b]} = a Here, U(f) is called Upper Riemann Integral & L(f) is called Lower Riemann Integral of f. bf(x) dx
