Mathematics

The key is to use the washer method with a horizontal axis of rotation that's outside the region. Since we're rotating about $y = -1$ (which lies below the region), each vertical slice at position $x$ sweeps out a washer whose outer radius is the distance from $y = -1$ to the curve $y = \sqrt{x}$, and whose inner radius is the distance from $y = -1$ to the $x$-axis. Set up the integral by finding these radii as functions of $x$, then integrate from $x = 0$ to $x = 4$.

The key is to use the washer method with a horizontal axis of rotation that's outside the region. Since we're rotating about $y = -1$ (which lies below the region), each vertical slice at position $x$ sweeps out a washer whose outer radius is the distance from $y = -1$ to the curve $y = \sqrt{x}$, and whose inner radius is the distance from $y = -1$ to the $x$-axis. Set up the integral by finding these radii as functions of $x$, then integrate from $x = 0$ to $x = 4$.