S spdrmncoo New member Joined Feb 27, 2006 Messages 20 Nov 26, 2006 #1 I would like some help with this problem, please. Find the volume bounded above by the sphere p = 6^(1/2) and below by the paraboloid z = x^2 + y^2. Locate the centroid of this region (x, y, z).
I would like some help with this problem, please. Find the volume bounded above by the sphere p = 6^(1/2) and below by the paraboloid z = x^2 + y^2. Locate the centroid of this region (x, y, z).
G galactus Super Moderator Staff member Joined Sep 28, 2005 Messages 7,203 Nov 26, 2006 #2 Using cylindrical coordinates: The equation of the sphere \(\displaystyle \L\\{\rho}=\sqrt{6}\) would be \(\displaystyle \L\\x^{2}+y^{2}+z^{2}=6\). Since \(\displaystyle \L\\x^{2}+y^{2}=r^{2}\) We have(I hope): \(\displaystyle \L\\\int_{0}^{2{\pi}}\int_{0}^{1}\int_{r^{2}}^{\sqrt{6-r^{2}}}rdzdrd{\theta}\)
Using cylindrical coordinates: The equation of the sphere \(\displaystyle \L\\{\rho}=\sqrt{6}\) would be \(\displaystyle \L\\x^{2}+y^{2}+z^{2}=6\). Since \(\displaystyle \L\\x^{2}+y^{2}=r^{2}\) We have(I hope): \(\displaystyle \L\\\int_{0}^{2{\pi}}\int_{0}^{1}\int_{r^{2}}^{\sqrt{6-r^{2}}}rdzdrd{\theta}\)
