1. **Problem Statement:** Set up the integral to calculate the volume of the region bounded by the cylinder $z = y^2$, the planes $x=0$, $x=1$, $y=-1$, $y=1$, and the XY-plane ($z=0$). 2. **Understanding the region:** The volume lies between the surface $z = y^2$ and the plane $z=0$ (XY-plane). 3. **Volume formula:** The volume under a surface $z = f(x,y)$ over a region $R$ in the $xy$-plane is given by: $$V = \iint_R f(x,y) \, dA$$ 4. **Region $R$ bounds:** Given $x$ ranges from 0 to 1 and $y$ ranges from -1 to 1. 5. **Setting up the integral:** Since $z = y^2$, the volume is: $$V = \int_{x=0}^1 \int_{y=-1}^1 y^2 \, dy \, dx$$ 6. **Explanation:** We integrate $y^2$ first with respect to $y$ over $[-1,1]$, then integrate the result with respect to $x$ over $[0,1]$. This integral setup will calculate the volume of the described region.