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The function provided for calculating the Riemann sum is:

\[
F(x) = \frac{6x^{9/2} + 3x^{7/2} + 3x^{5/2} + 9x^{3/2} - 9x^{1/2} + 3x^{3/2}}{3x^{3/2}}
\]

We need to simplify this function first. Dividing each term by \(3x^{3/2}\):

\[
F(x) = \frac{6x^{9/2}}{3x^{3/2}} + \frac{3x^{7/2}}{3x^{3/2}} + \frac{3x^{5/2}}{3x^{3/2}} + \frac{9x^{3/2}}{3x^{3/2}} - \frac{9x^{1/2}}{3x^{3/2}} + \frac{3x^{3/2}}{3x^{3/2}}
\]

Now simplifying each term:

\[
F(x) = 2x^3 + x^2 + x + 3 - 3x^{-1} + 1
\]

\[
F(x) = 2x^3 + x^2 + x + 4 - 3x^{-1}
\]

The next step is to compute the Riemann sum over the interval \([-1, 1]\). Since this function includes terms with both positive and negative exponents, this will involve a proper partitioning of the interval or using integration if required.

Would you prefer to proceed by calculating the Riemann sum for a given number of partitions, or would you like to use the definite integral to compute the area?

Here are five questions to consider:
1. Would you like me to demonstrate a specific method for calculating the Riemann sum?
2. Should we consider a certain number of intervals for the partitioning?
3. Do you want to see the calculation of the definite integral step by step?
4. Are there any specific points where you'd like more clarification on the simplification process?
5. Would you like to discuss the behavior of the function over the interval?

**Tip:** Simplifying complex functions before starting calculations can significantly reduce errors and make the process easier to follow.

Calcule el área por sumatoria de Riemann en el intervalo de -1 a 1 de la función F(x) = (6x^(9/2) + 3x^(7/2) + 3x^(5/2) + 9x^(3/2) - 9x^(1/2) + 3x^(3/2)) / (3x^(3/2))

Learn how to calculate the area using Riemann sum for the function F(x) = (6x^(9/2) + 3x^(7/2) + 3x^(5/2) + 9x^(3/2) - 9x^(1/2) + 3x^(3/2)) / (3x^(3/2)) in the interval from -1 to 1. The problem covers algebraic simplification and key calculus concepts, including the Riemann sum and definite integrals. Suitable for undergraduate calculus students.

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