1. **State the problem:** Evaluate the definite integral $$\int_1^3 x^2 \, dx$$. 2. **Recall the formula:** The integral of $$x^n$$ with respect to $$x$$ is $$\frac{x^{n+1}}{n+1} + C$$ for $$n \neq -1$$. 3. **Apply the formula:** For $$x^2$$, the antiderivative is $$\frac{x^{3}}{3}$$. 4. **Evaluate the definite integral:** $$\int_1^3 x^2 \, dx = \left[ \frac{x^3}{3} \right]_1^3 = \frac{3^3}{3} - \frac{1^3}{3}$$ 5. **Calculate the values:** $$= \frac{27}{3} - \frac{1}{3} = 9 - \frac{1}{3}$$ 6. **Simplify the result:** $$= \frac{27}{3} - \frac{1}{3} = \frac{26}{3} = 8 \frac{2}{3}$$ **Final answer:** $$\int_1^3 x^2 \, dx = \frac{26}{3} = 8 \frac{2}{3}$$.