If $xy = 15$ and $x + y = 11$, calculate the value of $x^3 + y^3$.
1. We start with the given equations:
   \[
   xy = 15 \quad \text{and} \quad x + y = 11
   \]

2. We need to find \(x^3 + y^3\). We use the identity for the sum of cubes:
   \[
   x^3 + y^3 = (x + y)(x^2 - xy + y^2)
   \]

3. First, we need to find \(x^2 + y^2\). We use the square of the sum of \(x\) and \(y\):
   \[
   (x + y)^2 = x^2 + y^2 + 2xy
   \]
   Substituting the known values:
   \[
   11^2 = x^2 + y^2 + 2 \cdot 15
   \]
   \[
   121 = x^2 + y^2 + 30
   \]
   Solving for \(x^2 + y^2\):
   \[
   x^2 + y^2 = 121 - 30 = 91
   \]

4. Now, we substitute \(x + y\), \(xy\), and \(x^2 + y^2\) into the identity for \(x^3 + y^3\):
   \[
   x^3 + y^3 = (x + y)(x^2 - xy + y^2)
   \]
   \[
   x^3 + y^3 = 11(x^2 - xy + y^2)
   \]
   \[
   x^3 + y^3 = 11(91 - 15)
   \]
   \[
   x^3 + y^3 = 11 \cdot 76
   \]
   \[
   x^3 + y^3 = 836
   \]

The final answer is \(\boxed{836}\)