We are given the equation:

$\frac{a}{b} + \frac{b}{a} = 1$

First, let's simplify the left-hand side:

$\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab}$

Thus, we now have the equation:

$\frac{a^2 + b^2}{ab} = 1$

Multiply both sides of this equation by $ab$:

$a^2 + b^2 = ab$

We are asked to find $a^3 + b^3$. We can use the identity for the sum of cubes:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Since we already know that $a^2 + b^2 = ab$, we can substitute $ab$ for $a^2 + b^2$:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2) = (a + b)(0) = 0$

Thus, $a^3 + b^3 = 0$.

Would you like further details on any part of this solution?

Here are 5 related questions:

1. What happens if $\frac{a}{b} + \frac{b}{a} \neq 1$?
2. How can we generalize this approach for other values of the sum?
3. Can we solve for specific values of $a$ and $b$?
4. What if $a$ and $b$ are complex numbers?
5. What is the geometric interpretation of this result?

Tip: Whenever you see an expression like $a^3 + b^3$, remember the identity $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$, which simplifies calculations.