Find volume of sphere cap with triple-integral

I'm trying to find the volume of the cap of a sphere with double/triple integral. However I keep getting the wrong answer. I have a sphere with a radius of one centered at the orgine and the cap is between z=1/2 and z=1.

Here's my integral:

$4\int\limits_{0}^{\sqrt{3}/2}\int\limits_{0}^{\sqrt{1-x^2}} \int\limits_{1/2}^{\sqrt{1-x^2-y^2}} dzdydx$

• Would it be easier to use spherical coordinates? – Nighty Feb 11 '15 at 13:48
• Hint: $\int \sqrt{a-x^2} dx = \frac12 (x \sqrt{a-x^2}+a \tan^{-1}(\frac{x}{\sqrt{a-x^2}}))+\text{constant}$ – Matthias Feb 11 '15 at 13:57
• I know how to calculate it and I've tried using the mathematica widget. However the value is contradictory of the value predicted by just using the wolfram mathworld formula for volume of a sphere cap. – user3517501 Feb 11 '15 at 15:06
• I don't want to know the answer in spherical coordinates but specifically why the integral above is not yielding .6544 – user3517501 Feb 11 '15 at 15:09