If f(x) ≥ 0 for all x in [a,b], then f(xi) ≥ 0 and Ʃf(xi*) ≥ 0.
and if all values for x between [a,b] and Δx = (b-a)/n then Δx ≥ 0 as well.
So, Ʃf(xi*))Δx ≥ 0. and if all values are ≥ 0, then the limit must be as well?
then ∫(a,b)f(x)dx = limn→∞Ʃf(xi*)Δx ≥ 0
ah k apparently have learned about that, found my notes on riemann sum, just didnt know it by name, from much ealier in the course. going to see what i can come up with with you guys' help so far and my notes :)
http://img42.imageshack.us/img42/5464/definitionofintegral.png
was the definition I found in my textbook, which i don't understand all that well, i can evaluate an integral but I am not very good an interpreting definitions like this :/
definition of an integral question...
if f(x) ≥ 0, how can you use the definition of an integral to prove that ∫(a,b)f(x)dx ≥ 0?
This seems like it is an easy question, and seems like one of those things that seems obvious but hard to explain, and the only definition of an integral I've been...
so here's my attempt:
http://img692.imageshack.us/img692/3284/graphed.png
with f(x) = |x| so f(t) = |t| graphed above, and the area from -1 to x would be
(1/2)t2 -1/2 = ∫(-1,x)f(t)dt, so
d/dx(∫(0,x)f(t)dt) = f(x)
d/dx(1/2x2) = |x|
x = |x|
that seem correct?
oh.. you that's kinda obvious now that you point it out :P thanks :)
ya that's basically what using the area is :P just didnt clue into what f(t) was :P
The Question
Let f(x) = |x|. use areas to evaluate ∫(-1,x)f(t)dt for all x. use this to show that d/dx∫(0,x)f(t)dt = f(x)
not sure hot to evaluate the integral using area when i don't know what f(t) is...
need to show that the integral is ≤ 14/3, so don't see how showing that its ≤ 2√3+2 which = 5.46 helps(?) or am i misunderstanding?
not sure what you mean by overestimating e-x
really not sure what's wanted for this question, not very good at this yet :/ I am just starting to understand the...
so find the maximum value of √(x+e-x) and multiply by the interval length of 3?
the function is increasing from [0,3] so max is at position 3, √(3+e-3 ≈ 1.7464, 1.7464x3 ≈ 5.2391, which is greater than 14/3 so guessing I'm doing something wrong for your suggestion?
gonna attempt the...
Homework Statement
Show that ∫(0,3) √(x+e^-x)dx ≤ 14/3 (hint: do not attempt to evaluate the integral)when looking at the integral from 0 to 3 on the graph I can see that this is true, but not sure how to go about showing this without evaluating the integral, any help as to how to go about...