Iff: [a,b] R is bounded and has only a finit number of discontinuous points, then f is Riemann integrable. 1
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![) Iff: [a,b] → R is bounded and has only a finit number of discontinuous points, then f is Riemann integrable.
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- If e is the unity in an integral domain D, prove that (e)a=a for all aD. [Type here][Type here]Prove the following theorem: If ƒ(x, y) is defined in an open region R of the xy-plane and if ƒx and ƒy are bounded on R, then ƒ(x, y) is continuous on R. (The assumption of boundedness is essential.)Prove that a contant ,x ,x^2 are integrable on [a,b]
- 3.1 Use the Cauchy integral formulas to evaluate the ∮c eizcos z/ z2(2z -3π) dz, where C is the region covered by circle |z - 2| = 3.Suppose the interval [2, 6] is partitioned into n = 4 subintervalswith grid points x0 = 2, x1 = 3, x2 = 4, x3 = 5, and x4 = 6.Write, but do not evaluate, the left, right, and midpoint Riemannsums for ƒ(x) = x2.Let be the boundary surface of the box enclosed by the planes x = 0, x = 2, y=0, y = 4, z =0 and z= 6 Approximate {{S e0.1xyz dS by using a Riemann sum asin Defi ni tion 1, taking the patches to be the rectanglesthat are the faces of the box and the points to be thecenters of the rectangles.