My Math Forum [Multivariable Calculus] Find the volume of the ellipsoid

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 May 20th, 2013, 09:25 PM #1 Newbie   Joined: May 2013 Posts: 2 Thanks: 0 [Multivariable Calculus] Find the volume of the ellipsoid Use multiple integrals, change of variables, spherical coordinates, et cetera to: Evaluate the volume of the ellipsoid ((x^2) / 16) + ((y^2) / 9) + ((z^2) / 25) = 1 Hint : Use the volume of sphere. I've worked through it, and have the limits of integration asS stands for integral symbol) SS z dA Where z=f(x,y) = (plus or minus) 5 *sqrt(1-(x2)/16 - (y2)) With (-3 *sqrt(1-(x2)/16) <= y <= 3 sqrt(1-(x2)/16)) And -4 <= x <= 4
 May 21st, 2013, 12:27 AM #2 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus Re: [Multivariable Calculus] Find the volume of the ellipsoi [color=#000000]$\frac{x^2}{16}+\frac{y^2}{9}+\frac{z^2}{25}=1$ $x=4r\sin(\phi)\cos(\theta),\;y=3r\sin(\phi)\sin(\t heta),\;z=5r\sin(\phi)$ and $\mathbb{d}x\;\mathbb{d}y\;\mathbb{d}z=3\cdot4\cdot 5\cdot r^2\sin(\phi)\mathbb{d}r\;\mathbb{d}\theta\;\mathb b{d}\phi=60 r^2\sin(\phi)\mathbb{d}r\;\mathbb{d}\theta\;\mathb b{d}\phi$ $V=8\cdot 60 \int_{0}^{1}\int_{0}^{\frac{\pi}{2}}\int_{0}^{\fra c{\pi}{2}}r^2\sin(\phi)\;\mathbb{d}\phi\;\mathbb{d }\theta\;\mathbb{d}r=480\int_{0}^{1}\int_{0}^{\fra c{\pi}{2}}r^2\;\mathbb{d}\theta\;\mathbb{d}r=240\p i\int_{0}^{1}r^2\;\mathbb{d}r=80\pi$ .[/color]
May 21st, 2013, 07:59 AM   #3
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Re: [Multivariable Calculus] Find the volume of the ellipsoi

Quote:
 Originally Posted by ZardoZ [color=#000000]$\frac{x^2}{16}+\frac{y^2}{9}+\frac{z^2}{25}=1$ $x=4r\sin(\phi)\cos(\theta),\;y=3r\sin(\phi)\sin(\t heta),\;z=5r\sin(\phi)$ and $\mathbb{d}x\;\mathbb{d}y\;\mathbb{d}z=3\cdot4\cdot 5\cdot r^2\sin(\phi)\mathbb{d}r\;\mathbb{d}\theta\;\mathb b{d}\phi=60 r^2\sin(\phi)\mathbb{d}r\;\mathbb{d}\theta\;\mathb b{d}\phi$ $V=8\cdot 60 \int_{0}^{1}\int_{0}^{\frac{\pi}{2}}\int_{0}^{\fra c{\pi}{2}}r^2\sin(\phi)\;\mathbb{d}\phi\;\mathbb{d }\theta\;\mathbb{d}r=480\int_{0}^{1}\int_{0}^{\fra c{\pi}{2}}r^2\;\mathbb{d}\theta\;\mathbb{d}r=240\p i\int_{0}^{1}r^2\;\mathbb{d}r=80\pi$ .[/color]
Where did you get the 8 from? Was it from bring the lower bounds of the limits of integration to 0? Also, how did you determine the limits of integration in this case?

 May 21st, 2013, 01:11 PM #4 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus Re: [Multivariable Calculus] Find the volume of the ellipsoi [color=#000000]Due to symmetry I computed the volume in the first octant and since a 3d cartesian system has 8 octants, I multiplied by 8.[/color]

 Tags calculus, ellipsoid, find, multivariable, volume

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# volume of ellpiseoid by double integration

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