To convert the given Riemann sum to an integral, we need to recognize the form of the sum and how it approximates the integral.

Given the expression:

$\lim_{n \to \infty} \left( \frac{7}{n} \sum_{i=1}^n 3^{-2 + \frac{7i}{n}} \right)$

Let's identify the components for conversion:

1. The term $\frac{7}{n}$ represents the width of each subinterval, $\Delta x$.
2. The summation $\sum_{i=1}^n$ represents the summation over the subintervals.
3. The term $3^{-2 + \frac{7i}{n}}$ represents the function evaluated at $x_i = -2 + \frac{7i}{n}$.

We recognize that this is a Riemann sum for the function $f(x) = 3^x$ over the interval $[-2, 5]$. This interval is derived from the limits of summation and the term $\frac{7i}{n}$, where the function is evaluated.

To convert this to an integral, we replace the sum with an integral and $\frac{7}{n}$ with $dx$:

$\int_{-2}^{5} 3^x \, dx$

Therefore, the given Riemann sum can be expressed as the integral:

$\lim_{n \to \infty} \left( \frac{7}{n} \sum_{i=1}^n 3^{-2 + \frac{7i}{n}} \right) = \int_{-2}^{5} 3^x \, dx$

Would you like me to solve this integral or provide more details? Here are some questions you might ask next:

1. How do you solve the integral $\int_{-2}^{5} 3^x \, dx$?
2. Can you explain Riemann sums in more detail?
3. What are some other examples of converting Riemann sums to integrals?
4. How can we verify the limits of integration in a Riemann sum?
5. What are the common mistakes to avoid when working with Riemann sums and integrals?

Tip: When converting Riemann sums to integrals, carefully identify the function and the limits of integration. This often involves recognizing the form of the sum and the intervals over which it is evaluated.