Can someone explain this summation definition to me?

[CookieSalesman]

A=limn→∞Rn=limn→∞[f(x1)Δx+f(x2)Δx+...+f(xn)Δx]

Consider the function f(x)=4√x, 1≤x≤16. Using the above definition, determine which of the following expressions represents the area under the graph of f as a limit.

I knew the correct answer was $$\sum \frac{15}{n} (4√x+\frac{15i}{n})$$

I figured out most of this, but the only thing I don't get is how you figure out is basically since you start the sum from i=1 to n, that you have to shift the sum up. If i=0, then I think there would be no "1+" term, right?
But let's say it was... i =5 to n. I have no idea why, or where you would put the added terms in order for the sum to work. I thought that it would be outside of the function 4√x but inside of the summnation, because, well, you're just adding values, right?
I don't see or understand intuitively why 1+ would go inside of 4√x.Similarly, I got this right but didn't understand the idea.
A=limn→∞Rn=limn→∞[f(x1)Δx+f(x2)Δx+...+f(xn)Δx]

Consider the function $$f(x) = \frac{ln(x)}{x}$$,3≤x≤10. Using the above definition, determine which of the following expressions represents the area under the graph of f as a limit.

Of course, the answer was Δx and $$\frac{ln 3+\frac{7i}{n}}{3+\frac{7i}{n}}$$, but just like above, wasn't sure why the $$\frac{ln3}{3}$$ went there. I thought if anything, it should be an added term, not mixed up with the main fraction.

 

[lurflurf]

The area is

$$\lim_{n\rightarrow\infty}\sum_{i=1}^n \dfrac{b-a}{n}\mathrm{f}\left(a+i\dfrac{b-a}{n}\right)$$

The function values are determined by dividing the interval into equal parts. a+i(b-a)/n are the points that that divide the interval evenly.

 

[CookieSalesman]

Thanks, I understand that, but just about the notation- Do you mean to say f(x)? And the X being all the stuff in parentheses?

But sorry, I have another confusion- if a is on the inside of those parentheses, isn't A the starting value? Then why, for example is the answer for that lnx/x one not the A value from, 3≤x≤10, which would make it 3, right? Instead of ln3?

 

[Mark44]

Thanks, I understand that, but just about the notation- Do you mean to say f(x)? And the X being all the stuff in parentheses?

I'm pretty sure he didn't mean f(x) or he would have written that. To understand what that notation means, try it out with a simple function like f(x) = x² and an interval [0, 2].

See what you get with, say, n = 4. The summation will have 4 terms.

But sorry, I have another confusion- if a is on the inside of those parentheses, isn't A the starting value?

a is the left endpoint of the interval.

Then why, for example is the answer for that lnx/x one not the A value from, 3≤x≤10, which would make it 3, right? Instead of ln3?

This doesn't make much sense, so I don't know what you're asking.

 

[vela]

But sorry, I have another confusion- if a is on the inside of those parentheses, isn't A the starting value? Then why, for example is the answer for that lnx/x one not the A value from, 3≤x≤10, which would make it 3, right? Instead of ln3?

It would have been clearer with parentheses. You should have written
$$\frac{\ln \left(3 + \frac{7i}n\right)}{3 + \frac{7i}n}$$ which I hope you recognize as $f\left(3 + \frac{7i}n\right)$ when $f(x) = \frac{\ln x}{x}$.