Find the volume bounded by the $xy$ plane, cylinder $x^2 + y^2 = 1$ and sphere $x^2 + y^2 +z^2 = 4$ 
Find the volume bounded by the $xy$ plane, cylinder $x^2 + y^2 = 1$ and sphere $x^2 + y^2 +z^2 = 4$.

I am struggling with setting up the bounds of integration. 
First, I will calculate the 'first-quadrant' piece of the volume. 
$z$  will traverse from $0$ to $2$. 
$x$ should start from the cylinder and go to the edge of the current circle of the sphere:
$$\sqrt{1-y^2} \le x \le \sqrt{4-x^2-z^2}$$
However, the same applies to $y$: (I am only calculating half of the volume right now, where the smaller circle is the lower bound): 
$$\sqrt{1-x^2} \le y \le \sqrt{4-y^2-z^2}$$ 
However, this cannot work as both $x$ and $y$ are dependent. 
What is the error?

Let $f(x, y) = \sqrt{4-(x^2 + y^2)}$ be the function which for each point closer than $4$ to the origin of the $xy$-plane gives the height of the top half of $x^2 + y^2 + z^2 = 4$ over that point. Then what you want is the integral of this function, over the unit disc.
Expressed in polar coordinates, this is given by
$$
\int_0^{2\pi}d\theta\int_0^1r\sqrt{4-r^2}dr
$$
