Using HallsofIvy's post, you need to use cylindrical co-ordinates to compute the integral and obtain the answer.
More formally, what you are doing is an integral transform by changing variables. You should have a change of variables theorem lying around telling you:
$$\iiint_D f(x, y, z) \space dV = \iiint_{D^\prime} f(r \text{cos}(\theta), r \text{sin}(\theta), z) \space J_{r, \theta}(x,y) \space dV' = \iiint_{D^\prime} f(r \text{cos}(\theta), r \text{sin}(\theta), z) \space J_{r, \theta}(x,y) \space dz d\theta dr$$
Where ##J_{r, \theta}(x,y)## is the Jacobian of the invertible transformation and ##dV' = dz d\theta dr## for cylindrical co-ordinates. For this particular problem:
$$J_{r, \theta}(x,y) = \left| \begin{array}{cc}
x_r & x_{\theta} \\
y_r & y_{\theta} \\
\end{array} \right| \quad \quad
x = r \text{cos}(\theta), y = r \text{sin}(\theta)$$
You have already found ##J_{r, \theta}(x,y) = r##. You have also found the limits for ##r## and ##\theta## already. What about the limits for ##z##? What does ##z = 2 + x^2## look like? It looks like it's bounded below by something.