Thread: Integration using Riemann sums

The question:
Suppose that

i) Show that f is increasing on the interval [0, 1/2].
ii) Find the upper Riemann sum for f with respect to the partition
{ } of [0, 1/2]
iii) Hence evaluate limit n -> infinity ( )

My attempt:
i) I took the derivative of f, and found the stationary points by equating it to 0. I noticed that the only stationary point was at 0 itself, thus the graph may only change gradient at this point. I substituted for 1/2 and found that the gradient was positive, so the interval [0, 1/2] is increasing.

ii) I worked out the width of each partition is , and the height of a given partition 'k' is . My sum then looked something like this:



This looks different to what's in part iii). Is it incorrect? Also, how do I solve part iii)? Thanks.

Any ideas?

Originally Posted by Glitch

The question:
Suppose that

i) Show that f is increasing on the interval [0, 1/2].
ii) Find the upper Riemann sum for f with respect to the partition
{ } of [0, 1/2]
iii) Hence evaluate limit n -> infinity ( )

My attempt:
i) I took the derivative of f, and found the stationary points by equating it to 0. I noticed that the only stationary point was at 0 itself, thus the graph may only change gradient at this point. I substituted for 1/2 and found that the gradient was positive, so the interval [0, 1/2] is increasing.

ii) I worked out the width of each partition is , and the height of a given partition 'k' is . My sum then looked something like this:

But the integral is

This looks different to what's in part iii). Is it incorrect? Also, how do I solve part iii)? Thanks.

Oh! I see what you did! I must have forgotten to multiply by width! >_<

Now that I understand that bit, how do I calculate the limit?

Well, you have now that the Riemann sum is and the limit of that, as n goes to infinity is the integral from 0 to 1/2.

But the limit as n goes to infinity is exactly the sum you want to find.

So, go ahead and integrate from 0 to 1/2: .
I recommend a trignometric substitution.

Oh, so I don't have to take a limit, I just solve it as an integral?