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I'm fairly new to this site so stick with me, I realize often times this is used by pretty educated people so I feel a bit out of place posting something like this on here. The problem in concern here is as follows: Riemann sum with integration problem

So I'm a bit lost on this and need some help on where I can maybe start this problem and maybe some clarification on what the answer is going to look like. Thanks!

For those who can't open or follow the link the problem is as follows: "Three terms of a left-hand sum used to approximate a definite integral $\int_a^b$ f(x) dx are as follows. (2+0* $\frac 43$)$^2$ * $\frac 43$ + (2 + 1*$\frac 43$)$^2$ * $\frac 43$ + (2 + 2 * $\frac 43$)$^2$ * $\frac 43$ Find possible values for 'a' and 'b' and a possible formula for f(x)"

Once again, I'll appreciate any help I can get, and I know I'm shooting way under the educational level of this site, but I thought this would be the best place to go for informed help. Thanks! (As of right now, I was able to find a and b, but I'm lost on being able to find the equation itself. I realize that the following can be converted into a form that may yield an equation: $$\lim\limits_{x \to \infty}\sum_{i=0}^n f(x_i^*)\delta x$$, but I'm fairly confused on how (if I'm right) this can be converted into an equation or how to find an equation here in general, again, all help is appreciated!)

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  • $\begingroup$ For elementary/homework/computation questions of this form, we generally ask that you show any progress you've made and you point out specifically where you are stuck. We want to help you understand the problem; not just solve it for you. By the way, don't feel bad asking questions of this sort on the site, many of the questions are of this level. Welcome to Math.SE! $\endgroup$ Commented Apr 24, 2020 at 23:11

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Hint
Look in your textbook. What is a three-term lefthand sum used to approximate the integral $\int_a^b f(x)\;dx$? Divide the interval $[a,b]$ into three parts, each of length $\Delta x = \frac{b-a}{3}$. The lefthand sum is $$ f(a+0\Delta x)\cdot\Delta x +f(a+1\Delta x)\cdot \Delta x +f(a+2\Delta x)\cdot\Delta x $$ Then look at your formula, and try to match it.

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  • $\begingroup$ Ok, thanks! I'm following what you're saying, so I was able to find a and b from your information, but I'm still lost on finding the actual function, I was thinking that I'd be able to find something after solving the sum but I don't know if that's right? $\endgroup$
    – joe
    Commented Apr 24, 2020 at 23:00

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