1. **Problem statement:** Sketch the region of integration for the integral $$\int_0^1 \int_0^{\sqrt{1+x^2}} f(x,y) \, dy \, dx$$ 2. **Understanding the limits:** - The outer integral has $x$ going from $0$ to $1$. - The inner integral has $y$ going from $0$ to $\sqrt{1+x^2}$. 3. **Interpreting the region:** - For each fixed $x$ in $[0,1]$, $y$ ranges from $0$ up to the curve $y = \sqrt{1+x^2}$. - The curve $y = \sqrt{1+x^2}$ is the upper boundary of the region. 4. **Sketching the region:** - Plot the curve $y = \sqrt{1+x^2}$ for $x$ in $[0,1]$. - The lower boundary is the $x$-axis ($y=0$). - The region is the set of points $(x,y)$ with $0 \leq x \leq 1$ and $0 \leq y \leq \sqrt{1+x^2}$. 5. **Summary:** - The region is bounded below by $y=0$, on the left by $x=0$, on the right by $x=1$, and above by $y=\sqrt{1+x^2}$. This completes the description and sketching of the region of integration.