- #1
Benny
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Hi, I'm having trouble setting up an integral for the following problem.
Q. Let D be the region inside the sphere x^2 + y^2 + z^2 = 4 in common with the region below the cone z = \frac{1}{{\sqrt 3 }}\sqrt {x^2 + y^2 }.
Using spherical coordinates find the mass of D if the mass density is z^2.
I keep on getting an answer which doesn't correspond to the given answer. I just need help setting up the integral. I get:
<br /> z = \frac{1}{{\sqrt 3 }}\sqrt {x^2 + y^2 } \Rightarrow \cos \phi = \frac{1}{{\sqrt 3 }}\sin \phi \Rightarrow \phi = \frac{\pi }{3}<br />
<br /> x^2 + y^2 + z^2 = 4 \Rightarrow \rho = 2<br />
<br /> m = \int\limits_{}^{} {\int\limits_{}^{} {\int\limits_E^{} {z^2 dV} } } <br />
<br /> = \int\limits_{}^{} {\int\limits_{}^{} {\int\limits_E^{} {\rho ^2 \cos ^2 \left( \phi \right)dV} } } <br />
<br /> = \int\limits_{\frac{\pi }{3}}^\pi {\int\limits_0^{2\pi } {\int\limits_0^2 {\rho ^2 \cos ^2 \left( \phi \right)} } } \rho ^2 \sin \left( \phi \right)d\rho d\theta d\phi <br />
<br /> = \int\limits_{\frac{\pi }{3}}^\pi {\int\limits_0^{2\pi } {\int\limits_0^2 {\rho ^4 \cos ^2 \left( \phi \right)} } } \sin \left( \phi \right)d\rho d\theta d\phi <br />
The evaluation of the integral is fairly straight forward. It's just setting up the correct integral which is giving me problems. Can someone go through my working and show me where I stuffed up?
Q. Let D be the region inside the sphere x^2 + y^2 + z^2 = 4 in common with the region below the cone z = \frac{1}{{\sqrt 3 }}\sqrt {x^2 + y^2 }.
Using spherical coordinates find the mass of D if the mass density is z^2.
I keep on getting an answer which doesn't correspond to the given answer. I just need help setting up the integral. I get:
<br /> z = \frac{1}{{\sqrt 3 }}\sqrt {x^2 + y^2 } \Rightarrow \cos \phi = \frac{1}{{\sqrt 3 }}\sin \phi \Rightarrow \phi = \frac{\pi }{3}<br />
<br /> x^2 + y^2 + z^2 = 4 \Rightarrow \rho = 2<br />
<br /> m = \int\limits_{}^{} {\int\limits_{}^{} {\int\limits_E^{} {z^2 dV} } } <br />
<br /> = \int\limits_{}^{} {\int\limits_{}^{} {\int\limits_E^{} {\rho ^2 \cos ^2 \left( \phi \right)dV} } } <br />
<br /> = \int\limits_{\frac{\pi }{3}}^\pi {\int\limits_0^{2\pi } {\int\limits_0^2 {\rho ^2 \cos ^2 \left( \phi \right)} } } \rho ^2 \sin \left( \phi \right)d\rho d\theta d\phi <br />
<br /> = \int\limits_{\frac{\pi }{3}}^\pi {\int\limits_0^{2\pi } {\int\limits_0^2 {\rho ^4 \cos ^2 \left( \phi \right)} } } \sin \left( \phi \right)d\rho d\theta d\phi <br />
The evaluation of the integral is fairly straight forward. It's just setting up the correct integral which is giving me problems. Can someone go through my working and show me where I stuffed up?
