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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 $\displaystyle \varphi$, it should be $\displaystyle \frac{\pi}{6}$ not $\displaystyle \frac{\pi}{3}$.

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) = $\displaystyle 4-\sqrt{3(0^2+2^2)}=4-.535898=3.4641$

Cone: V=$\displaystyle (\frac{1}{3}\pi*2^2*3.4641)=14.5104$

Sphere: V=$\displaystyle (\frac{4}{3}\pi*2^3)=\frac{16\pi}{3}$

Cylinder: V=$\displaystyle (\pi*2^2*.535898)=6.7343$

Total volume = Cone + Cylinder - Sphere = 14.5104 -$\displaystyle \frac{16\pi}{3}$ + 6.7343 = 4,48953

Since I only need from $\displaystyle \frac{\pi}{4}$ to $\displaystyle \frac{\pi}{2}$ (one eight) and not 2$\displaystyle \pi$, I divide by 8.

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

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

For the spherical coordinates case, your limits for $\displaystyle \rho$ look wrong. Firstly, if you're above the sphere, the lower limit should be 2. Secondly, you have not converted the Cartesian equation for the cone correctly into sphericals. The equation $\displaystyle \phi =\pi/3$ is a cone with vertex at the origin; this is not what you were given.