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This problem asks to express the given Riemann sum as a definite integral. Let's break it down.

The sum provided is:
\[
\lim_{n \to \infty} \sum_{i=1}^{n} \left[ 6(x_i^*)^3 - 5x_i^* \right] \Delta x
\]

This is a Riemann sum for the function \( f(x) = 6x^3 - 5x \) over the interval \([2, 7]\).

### Step-by-Step Solution:

1. The term \( x_i^* \) represents the sample points in the interval \([2, 7]\), and as \( n \to \infty \), the sum becomes the integral over that interval.
2. \( \Delta x \) represents the width of each subinterval, which corresponds to \( dx \) in the integral.
3. The Riemann sum converges to the integral of \( f(x) = 6x^3 - 5x \) over the interval \([2, 7]\).

Thus, the limit of the sum is expressed as the definite integral:
\[
\int_{2}^{7} \left( 6x^3 - 5x \right) dx
\]

Would you like me to solve this integral for you or provide further clarification on any step?

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Here are some related questions:
1. How do you evaluate a definite integral of a polynomial function?
2. Can Riemann sums be used for functions with more complex expressions?
3. What is the geometric interpretation of a definite integral?
4. How do changes in the interval affect the value of the integral?
5. How do you calculate the error in Riemann sum approximations?

**Tip:** When converting a Riemann sum to a definite integral, always identify the function and the interval first!

Express the limit of the sum \(\lim_{n \to \infty} \sum_{i=1}^{n} \left[ 6(x_i^*)^3 - 5x_i^* \right] \Delta x\) as a definite integral on the interval [2, 7].

Learn how to convert a Riemann sum into a definite integral over the interval [2, 7]. The function in the sum is 6x^3 - 5x. This problem illustrates key calculus concepts, including the Riemann sum and the Fundamental Theorem of Calculus. Suitable for advanced high school or introductory college-level students.

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