
COS X Objective is to prove that log' x is improperly integrable on le,c).
Is the function x2 cos(x3) improperly integrable on (0,0)? Explain.
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
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]....
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a < x < b.)
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a
B1) Prove that the function f(x,y) = c=y 0 otherwise, is integrable over [0, 1] x [0, 1].
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...
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.
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
#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...
Prove the following: where B JA, and X is an integrable random variable