1. **Problem:** Sketch the region of integration for the integral $$\int_1^{10} \int_0^{\frac{1}{y}} y e^{xy} \, dx \, dy$$ and evaluate it. 2. **Understanding the region:** The limits for $y$ are from 1 to 10. 3. For each fixed $y$, $x$ varies from 0 to $\frac{1}{y}$. 4. This means the region is bounded by: - $y$ between 1 and 10, - $x$ between 0 and $\frac{1}{y}$. 5. To sketch: - Plot the curve $x = \frac{1}{y}$. - The region lies between $y=1$ and $y=10$ vertically, - and between $x=0$ and $x=\frac{1}{y}$ horizontally. 6. The curve $x=\frac{1}{y}$ is a hyperbola in the $xy$-plane. 7. **Evaluation:** $$\int_1^{10} \int_0^{\frac{1}{y}} y e^{xy} \, dx \, dy = \int_1^{10} y \left[ \frac{e^{xy}}{y} \right]_0^{\frac{1}{y}} \, dy = \int_1^{10} (e^1 - 1) \, dy = (e - 1) \int_1^{10} 1 \, dy = (e - 1)(10 - 1) = 9(e - 1)$$ 8. **Final answer:** $$9(e - 1)$$ This completes the sketch and evaluation of the first integral. Note: The user asked for sketches of regions for multiple questions but per instructions, only the first problem is solved and sketched here.