1. **State the problem:** We are given that $f$ is continuous and $$\int_0^9 f(x) \, dx = 10,$$ and we need to evaluate $$\int_0^3 x f(x^2) \, dx.$$ 2. **Use substitution to simplify the integral:** Let $t = x^2$. Then, $$dt = 2x \, dx \implies x \, dx = \frac{dt}{2}.$$ 3. **Change the limits of integration:** When $x=0$, $t=0^2=0$. When $x=3$, $t=3^2=9$. 4. **Rewrite the integral:** $$\int_0^3 x f(x^2) \, dx = \int_0^9 f(t) \frac{dt}{2} = \frac{1}{2} \int_0^9 f(t) \, dt.$$ 5. **Use the given integral value:** $$\frac{1}{2} \int_0^9 f(t) \, dt = \frac{1}{2} \times 10 = 5.$$ **Final answer:** $$\boxed{5}.$$