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:

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Attempted solution:

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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:

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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 $\displaystyle f(t) = \sqrt{1+t^3}$ over the interval $\displaystyle (0,x^2)$. But, there is not really any simplification you can do.

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

$\displaystyle \lim_{n\to \infty} \sum_{i = 1}^n f(x_i^*)\Delta x$

In general, $\displaystyle \Delta x = \dfrac{b - a}{n}$ and $\displaystyle x_i = a + i\Delta x$.

So, instead of factoring out $\displaystyle \sqrt{\dfrac{1}{n^3}}$, you should only factor out $\displaystyle \Delta x$. So, you want $\displaystyle \sqrt{\dfrac{i}{n^3}} = \left(\sqrt{a + i\Delta x}\right)\Delta x$. The hint is to factor out $\displaystyle \dfrac{1}{n}$. So, that must be $\displaystyle \Delta x$:

$\displaystyle \Delta x = \dfrac{b-a}{n} = \dfrac{1}{n}$

That means $\displaystyle b-a = 1$. Since the square root does not have a sum inside, it appears that $\displaystyle a=0, b=1$. Can you figure out the rest?

Re: Two problems involving FDC/Riemman sums