4 Attachment(s)

Two problems involving FDC/Riemman sums

I have two questions I need help with. I have attempted both of them but I have no idea how to proceed from here.

Question 1:

Attachment 29640

Attempted solution:

Attachment 29641

I used product rule then I solved for d/dx of the integral but I have no idea how to solve for the integral in the first term of my final equation.

Question 2:

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Attempted solution:

Attachment 29643

The question gave a hint: "Rewrite it as lim_n-> infinity 1/n(*) then relate what you get to a Riemann sum. I got it to the point where I got the 1/n term out of the sum but I have no idea what the formula for the sum of a square root is and I can't find it anywhere.

I'm not looking for the full solution I just want to know if I did so far is correct and how to continue further.

Thanks in advance

Re: Two problems involving FDC/Riemman sums

For the first one, you have the average value of the function over the interval . But, there is not really any simplification you can do.

For the second one, a Riemann sum looks like this:

In general, and .

So, instead of factoring out , you should only factor out . So, you want . The hint is to factor out . So, that must be :

That means . Since the square root does not have a sum inside, it appears that . Can you figure out the rest?

Re: Two problems involving FDC/Riemman sums