I've reviewed the assignment you uploaded. To help with the specific questions, I will address each problem one by one and provide the solutions accordingly.

I'll start by calculating the answers based on the problems listed in your file. For clarity, I will number my answers according to the corresponding questions in the paper.

I'll begin with Problem 1 and proceed sequentially. Let me start with Problem 1 first:

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Problem 1:

The given function is $f(x) = \frac{x^2}{4} + 3$, and we need to calculate the Riemann sum for the integral over the interval $[0, 3]$.

We will use the Riemann sum formula:

$R_n = \sum_{i=1}^{n} f(x_i) \Delta x$

Where $\Delta x = \frac{3-0}{n} = \frac{3}{n}$, and the sample points are at the right-hand endpoints. The right-hand endpoints of each sub-interval are given by:

$x_i = \frac{3i}{n}$

Thus, the Riemann sum $R_n$ can be written as:

$R_n = \sum_{i=1}^{n} \left( \frac{\left(\frac{3i}{n}\right)^2}{4} + 3 \right) \cdot \frac{3}{n}$

Expanding the terms inside the summation:

$R_n = \sum_{i=1}^{n} \left( \frac{9i^2}{4n^2} + 3 \right) \cdot \frac{3}{n}$

Simplifying:

$R_n = \sum_{i=1}^{n} \left( \frac{27i^2}{4n^3} + \frac{9}{n} \right)$

Thus, the Riemann sum $R_n$ for the given function and interval is:

$R_n = \frac{27}{4n^3} \sum_{i=1}^{n} i^2 + \frac{9}{n} \sum_{i=1}^{n} 1$

This is the general form for the Riemann sum for this problem.

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I will continue to work through the next problems in the same manner, detailing each calculation. Would you like me to continue with the next questions or do you need further details on Problem 1?