- #1
perlawin
- 3
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1. True or false (show work): For all bounded functions:
(L) ∫_a^b▒f(x)dx≤∫_a^b▒f(x)dx≤(U)∫_a^b▒f(x)dx
2. (L) ∫_a^b▒f(x)dx= sup{L(f,P) s.t P is a partition of [a,b]}
(U)∫_a^b▒f(x)dx= inf{U(f,P) s.t. P is a partition of [a,b]}
3. I am sure that this is true. What I want to do is prove it by induction. Specifically, prove that the first inequality holds and then show that the second one does. I have drawn pictures representing a base case (how the lower integral is less than the regular one), and I have pictures that illustrate how the amount of area increases but is never exact. How do I actually write it out?
(L) ∫_a^b▒f(x)dx≤∫_a^b▒f(x)dx≤(U)∫_a^b▒f(x)dx
2. (L) ∫_a^b▒f(x)dx= sup{L(f,P) s.t P is a partition of [a,b]}
(U)∫_a^b▒f(x)dx= inf{U(f,P) s.t. P is a partition of [a,b]}
3. I am sure that this is true. What I want to do is prove it by induction. Specifically, prove that the first inequality holds and then show that the second one does. I have drawn pictures representing a base case (how the lower integral is less than the regular one), and I have pictures that illustrate how the amount of area increases but is never exact. How do I actually write it out?