1. **Problem statement:** We want to change the order of integration for the integral $$\int_0^4 \int_0^y f(x,y) \, dx \, dy$$ where the region of integration is bounded by $x=0$, $y=4$, and $y=x$. 2. **Understanding the region:** The original integral integrates $x$ from $0$ to $y$, and then $y$ from $0$ to $4$. This means the region is all points $(x,y)$ such that: $$0 \leq x \leq y \leq 4$$ 3. **Changing the order of integration:** To switch the order, we describe the region in terms of $y$ first, then $x$. For a fixed $x$, $y$ ranges from $y=x$ (the line) up to $y=4$ (the horizontal line). Also, $x$ ranges from $0$ to $4$. 4. **New integral limits:** $$\int_0^4 \int_x^4 f(x,y) \, dy \, dx$$ 5. **Summary:** The integral with reversed order is $$\int_0^4 \int_x^4 f(x,y) \, dy \, dx$$ This correctly describes the same triangular region and allows integration first with respect to $y$, then $x$.