1. **Problem a)**: Evaluate or describe the region of integration for $$\int_0^4 \int_{\frac{4-y}{7}}^1 f(x,y) \, dx \, dy.$$ - Here, $y$ ranges from 0 to 4. - For each fixed $y$, $x$ ranges from $\frac{4-y}{7}$ to 1. 2. **Problem б)**: Evaluate or describe the region of integration for $$\int_{-\sqrt{2}}^{-1} \int_{-2 - x^2}^0 f(x,y) \, dy \, dx + \int_{-1}^0 \int_x^0 f(x,y) \, dy \, dx.$$ - The first integral covers $x$ from $-\sqrt{2}$ to $-1$, and for each $x$, $y$ ranges from $-2 - x^2$ to 0. - The second integral covers $x$ from $-1$ to 0, and for each $x$, $y$ ranges from $x$ to 0. 3. **Explanation of regions:** - For a), the region is bounded below by the curve $x = \frac{4-y}{7}$ and above by $x=1$, with $y$ between 0 and 4. - For б), the region is split into two parts: - Part 1: $x \in [-\sqrt{2}, -1]$, $y \in [-2 - x^2, 0]$ (a curved lower boundary). - Part 2: $x \in [-1, 0]$, $y \in [x, 0]$ (a linear lower boundary). 4. **Summary:** - These integrals define integration over specific regions in the $xy$-plane. - Without a specific function $f(x,y)$, the integrals represent the total integral over these regions. Final answer: The integrals describe the integration of $f(x,y)$ over the specified regions in the $xy$-plane as detailed above.