Sketch the region, the solid, and a typical disk/washer/shell (your choice) $y=x^2, y=1$; about the line $y=-1$
Alright so I have the sketch drawn but I cannot figure out if I'm doing this correctly because the $y=-1$ is throwing me off. The answer I got is $4/5\pi$.
I got that answer by using horizontal slices $R=\sqrt{y}=x$ and $r=y$. 
$$
\begin{align}
V&=2\pi \int_0^1 y(\sqrt{y})= \int_0^1(y)^{3/2}\\
&=2\pi \frac{2}{5}y^{5/2} \bigg|_0^1=\frac{4}{5}\pi
\end{align}
$$
Help if I went about this the wrong way. This stuff is just completely confusing.
 A: The region being rotated is not hard to sketch. The parabola $y=x^2$ meets the line $y=1$ at $x=\pm 1$. The region is symmetric about the $y$ axis. We take advantage of that symmetry. So we will find the volume when the part from $x=0$ to $x=1$ is rotated, and then double the result. 
You can probably see that the solid has a hole in it. The solid is in fact a cylinder with a hole drilled in it. The hole is quite wide at the two ends, and narrower in the middle.  
Take a typical cross-section "at" $x$. The cross-section is a circle with a circular hole in it. 
The radius of the outer circle is $2$. The radius of the inner circle is $x^2-(-1)$, though I visualize it as $x^2+1$. 
So the area of cross-section at $x$ is $\pi(2^2-(x^2+1)^2)$. 
Integrate from $x=0$ to $x=1$, then double.
Remark: Alternately, we could find the volume of the cylinder by basic geometry, and concentrate on finding the volume of the hole.
For the volume of the hole, if you are initially uncomfortable at rotating about the line $y=-1$, and prefer to rotate about the $x$-axis, you could raise everything by $1$. So then we would be dealing with the region below the line $y=2$ and above the curve $y=x^2+1$, rotated about the $x$-axis.  
A: You could also use the shell method to work this problem by finding an integral similar to the one you have: You could find $\int_0^1 2\pi r(y)h(y) dy$ where
$r(y)=y+1$, the distance from a typical horizontal line segment to the axis of revolution, and $h(y)=2\sqrt y$, the length of a typical horizontal line segment.  
