Confused on double integral in polar cords

[Samuelb88]

Homework Statement

Use polar coordinates to find the volume of the solid enclosed by the hyperboloid -x^2-y^2+z^2=1 and the plane z=2.

The Attempt at a Solution

Solving for z of the equation of the hyperboloid I find z = Sqrt(1 + x^2 + y^2). Letting z = 2 to determine the curve of intersection I find that 3 = x^2 + y^2, or r = Sqrt(3). Thus:

$$\int _{0}^{2Pi} \int _{0}^{3^(^1^/^2^)} (1+r^2)^(^1^/^2^)rdrd\theta$$

Making the substitution u = 1 + r^2 gives:

$$\frac{1}{3}\right) \int _{0}^{2Pi} 7d\theta = \frac{14}{3} Pi$$

The back of my book has 4/3*Pi. I don't understand how I am doing this problem wrong.

 

[LCKurtz]

Your integrand should be z_upper - z_lower.