The double integral to evaluate is:

$\int_{0}^{11} \int_{1}^{e^x} \frac{x}{y} \, dy \, dx$

Step-by-Step Solution:

1. Inner Integral (with respect to $y$):

   The inner integral is:

   $\int_{1}^{e^x} \frac{x}{y} \, dy$

   Since $\frac{x}{y}$ is a simple function in $y$, we can integrate it easily:

   $\int \frac{x}{y} \, dy = x \ln(y)$

   Now, apply the limits of integration $y = 1$ to $y = e^x$:

   $x \ln(e^x) - x \ln(1)$

   We know $\ln(e^x) = x$ and $\ln(1) = 0$, so this simplifies to:

   $x \cdot x - x \cdot 0 = x^2$

   Therefore, the result of the inner integral is $x^2$.

2. Outer Integral (with respect to $x$):

   Now, the outer integral becomes:

   $\int_{0}^{11} x^2 \, dx$

   The integral of $x^2$ is:

   $\frac{x^3}{3}$

   Apply the limits $x = 0$ to $x = 11$:

   $\frac{11^3}{3} - \frac{0^3}{3} = \frac{1331}{3}$

   Thus, the value of the double integral is:

   $\boxed{\frac{1331}{3}}$

Let me know if you'd like more details or have any questions! Here are some related questions to explore:

1. How would the result change if the limits of integration for $y$ were different?
2. What is the significance of logarithmic functions in integration?
3. How does changing the exponent in $e^x$ affect the double integral?
4. Can this integral be solved using polar coordinates?
5. How would you approach solving a triple integral involving a similar function?

Tip: When working with nested integrals, always handle one variable at a time, simplifying before proceeding to the next integration.