To express the given limit as a definite integral, we recognize that the sum represents a Riemann sum for the integral. In this case, the function being integrated is:

$f(x) = \frac{9}{x^7}$

The interval of integration is $[1, 8]$, as specified. Therefore, we can rewrite the limit as a definite integral:

$\int_{1}^{8} \frac{9}{x^7} \, dx$

This expression is the definite integral form of the given limit.

Would you like a detailed step-by-step explanation of how I arrived at this answer, or do you have any specific questions?

Related Questions:

1. How do you identify the function within a Riemann sum?
2. What are the steps to convert a Riemann sum into a definite integral?
3. How would this process change if the function had a different exponent?
4. What is the importance of the interval $[a, b]$ in defining the definite integral?
5. How do you interpret the terms $\Delta x_i$ and $c_i$ in a Riemann sum?

Tip:

When converting Riemann sums to integrals, carefully identify the function inside the sum and the interval over which you’re integrating. This helps in setting up the correct integral bounds and integrand.