1. **State the problem:** Calculate the definite integral $$\int_2^9 x \, dx$$. 2. **Formula and rules:** The integral of $$x$$ with respect to $$x$$ is given by $$\int x \, dx = \frac{x^2}{2} + C$$, where $$C$$ is the constant of integration. For definite integrals, we evaluate the antiderivative at the upper and lower limits and subtract. 3. **Apply the formula:** $$\int_2^9 x \, dx = \left[ \frac{x^2}{2} \right]_2^9 = \frac{9^2}{2} - \frac{2^2}{2}$$ 4. **Calculate values:** $$= \frac{81}{2} - \frac{4}{2} = \frac{81 - 4}{2} = \frac{77}{2}$$ 5. **Final answer:** $$\int_2^9 x \, dx = \frac{77}{2} = 38.5$$ This means the area under the curve $$y = x$$ from $$x=2$$ to $$x=9$$ is $$38.5$$.