Not Able To Obtain The Same Answer In Spherical/Cylindrical Coordinate

Hi,

I have to solve this problem using spherical coordinate and validate my answer using cylindrical coordinate, but I have 2 different answers, obviously!

Can anyone tell me what I am doing wrong? (See attached file for the problem and what I did)

Thanks, Mike.

EDIT: sorry, I used the wrong angle for the limit of $\varphi$ , it should be $\frac{\pi}{6}$ not $\frac{\pi}{3}$ .

My new answer is 0,420894

EDIT 2: I drew the solid in my cad software and came up with a volume of 0,561.

I tried to do it the long way, calculating the volume of the cone, sphere and cylinder.

H of cone (z) = $4-\sqrt{3(0^2+2^2)}=4-.535898=3.4641$

Cone: V= $(\frac{1}{3}\pi*2^2*3.4641)=14.5104$
Sphere: V= $(\frac{4}{3}\pi*2^3)=\frac{16\pi}{3}$
Cylinder: V= $(\pi*2^2*.535898)=6.7343$

Total volume = Cone + Cylinder - Sphere = 14.5104 - $\frac{16\pi}{3}$ + 6.7343 = 4,48953
Since I only need from $\frac{\pi}{4}$ to $\frac{\pi}{2}$ (one eight) and not 2 $\pi$ , I divide by 8.

$\frac{4,48953}{8}=0,561191$