If I have a limit going to infinity with Σ with n on top and i=1 on bottom that has (4/n) √(8+(4i/n)) inside and I know that A=0 and B=4 for the upper and lower limit when changing this into an integral. So what is f(x)? Why wouldn't it just be √x?
∑(4/n)√(8+(4k)/n)
This is supposed to be ∑ f(xi) ∆x
so you want
∫[0..4] √(8+x) dx
But what would represent f(x) in that circumstance? Like what is the function that has xi plugged into it?
huh? I showed you the integral, which is f(x) dx
If you go to some math web site, you will find that the sum and the integral are the same value.
xi = i (4/n)
I just used k, since most math sites interpret i as the imaginary number root(-1)
To find the function f(x), we need to evaluate the given summation expression and convert it into an integral. Let's break down the steps:
1. Start by rewriting the given summation with Σ notation:
2. Notice that the term inside the summation is (4/n) √(8 + (4i/n)). By rewriting i/n as x, we can convert the summation into an integral. So, let's make this substitution:
i/n = x ⇒ i = nx.
3. Perform the substitution in the summation expression:
4. Now, we can express the summation as an integral. Recall that as n approaches infinity, the width of the interval (b - a) decreases. Since we are given A = 0 and B = 4 as the limits, the integral becomes:
5. Simplify the integrand inside the integral:
6. Finally, integrate the expression:
Therefore, the function f(x) is:
Now, coming to your question about why f(x) wouldn't just be √x: The function inside the integral (√(8 + 4x)) is different from √x. Although they may appear similar, they are fundamentally different. In this case, the expression inside the square root depends on the variable 'x' and has a constant term. Thus, it cannot be simplified to √x.