Mathematics

When you revolve a region around a horizontal line that's not the $x$-axis, the key is to shift your perspective so the axis of rotation becomes your reference point. Here, revolving around $y = -1$ means every point in $T$ is actually a distance of (its $y$-coordinate plus 1) away from the axis. Use the washer method with outer radius $R(x) = x^2 - (-1) = x^2 + 1$ and inner radius $r(x) = 0 - (-1) = 1$, then integrate the area of each washer from $x = 0$ to $x = 2$.

When you revolve a region around a horizontal line that's not the $x$-axis, the key is to shift your perspective so the axis of rotation becomes your reference point. Here, revolving around $y = -1$ means every point in $T$ is actually a distance of (its $y$-coordinate plus 1) away from the axis. Use the washer method with outer radius $R(x) = x^2 - (-1) = x^2 + 1$ and inner radius $r(x) = 0 - (-1) = 1$, then integrate the area of each washer from $x = 0$ to $x = 2$.