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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.

Evaluate the double integral: ∫₀¹ ∫₁ᵉˣ (x/y) dy dx

Learn how to evaluate the double integral ∫₀¹ ∫₁ᵉˣ (x/y) dy dx with a step-by-step guide. This solution involves logarithmic integration and the application of fundamental calculus concepts. Suitable for university-level students.

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