Math 409-502Harold P. [email protected] vocabulary• partition• mesh• refinement• upper sum• lower sum• Riemann sum• integrable functionSome theorems• Bounded monotonic functions are integrable.• Bounded continuous functions are integrable.• Integration and differentiation are inverse operations (fundamental theorem of calculus).Math 409-502 November 17, 2004 — slide #2Partition and meshA partition of a compact interval [a, b] is a subdivision of the interval.A compact interval [1, 5]1 5A partition of the interval: division points 1, 2, 4, 51 2 4 5Symbolic notation for the division pointsx0x1x2x3The mesh of a partition is the maximum width of the subintervals. In the above example, themesh is 4 − 2 = 2.Math 409-502 November 17, 2004 — slide #3Upper sumThe upper sum of a bounded function for a partition of a compact interval means the sum overthe subintervals of the supremum of the function on the subinterval times the width of thesubinterval.x0x1x2x3y = f(x)Symbolic notation:n∑j= 1(xj− xj−1) sup[xj−1, xj]f (x).Math 409-502 November 17, 2004 — slide #4Lower sumThe lower sum is defined similarly with the infimum in place of the supremum.x0x1x2x3y = f(x)Symbolic notation:n∑j= 1(xj− xj−1) inf[xj−1, xj]f (x).Math 409-502 November 17, 2004 — slide #5Integrable functionsA function defined on a compact interval [a, b] is integrable if (i) the function is bounded, and(ii) for every e > 0, there exists δ > 0 such that for every partition of mesh < δ the upper sumfor the partition and the lower sum for the partition differ by less than e.Example. A constant function is integrable because every upper sum equals every lower sum.Example. f (x) =(1, if x is rational0, if x is irrationalis not integrable because every lower sum equals 0, but every upper sum equals the width ofthe interval.Math 409-502 November 17, 2004 — slide #6Homework• Read sections 18.1 and 18.2, pages 241–244.• Consider the integrable function f (x) = x on the interval [1, 2]. How small must the meshof a partition be in order to guarantee that the upper sum and the lower sum differ by lessthan 1/10?• Do exercise 18.2/3 on page 248.Math 409-502 November 17, 2004 — slide