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Rieman Integral and Rieman sums Exercise 6. (a) Prove that if f is integrable on [0,...
6. (25) Is the function f(x) below integrable on the interval (0, 3)? Prove your answer using upper sums and lower sums, and if f is integrable, find Sof by computing L(f) or U(f) directly. f(x) = 0 <0 x +1 0 < x < 1 x=1 2 > 1 I 1 Ž
Three questions!please!
7. Prove that J(x) is integrable on (0,b), and calculate their integral. 8. Prove that the following function is integrable on [0, 1], and calculate the integral. 1 if for some n E N 0 (z)= otherwise. 8. Prove that if f is integrable on (a, b, then f2 is also integrable on la,b
Consider the function f: [-1,1] defined by f(x)= 1, if x<0; 2016, if x=0; 1908, if x>0 Prove that f is integrable on [-1,1] (by using partitions and upper/lower sums)
1. f : riemann integrabel in [0,inf). prove that f is lebesgue integrable iff the improper integral converges absolutly.
hint
This exercise 5 to use the definition of Riemann integral
F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if С > 0, then, is also integrable on [a,b, (6 Marks) (2) If C 0, i, still integrable (assuming f(x) 0 for any x E [aA)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval...
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if C>0, then 7 is also integrable on la,b] (6 Marks) (2) If C 0, i, still integrable (assuming f(x)关0 for any x E [aM)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval [a, b]....
Prove that if ? is integrable on [?, ?] and ?(?) ≥ 0 for all ?
in [?, ?], then
[ f(x)dx > 0 7. Prove that if f is integrable on [a, b] and f(x) > 0 for all x in [a, b], then sof(x)dx > 0.
exercice 6
6. The goal of this problem is to prove that a function is Riemann integrable if and only if its set of discontinuities has measure 0. So, assume f: a, bR is a bounded function. Define the oscillation of f at , w(f:z) by and for e >0 let Consider the following claims: i- Show that the limit in the definition of the oscillation always exists and that f is continuous at a if and only if w(f;...
#4
(4) Use the Box-sum criterion to prove that if f is integrable on [a, b] and is also integrable on |b,e, then f is integrable on la, e) and Je fdr- o fdz+ (5) Suppose that (r) 2 0 and is continuous on a, b). Prove that if f - 0, then f(x) = 0 for all x E a,b]. Hint: Assume to the contrary that there is some r E [a, b] where f(x) > 0. What can...