Mathematics: Find the volume of the ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1

T = 0.9 (6 feet) = 5.4 feet. I think hoped It helped...

Find the volume of the ellipsoid x 2 a 2 + y 2 b

Ответ:

janakibubbles7711

14.03.2022

Use a modified version of spherical coordinates.

$x(\rho,\theta,\varphi)=a\rho\cos\theta\sin\varphi$
$y(\rho,\theta,\varphi)=b\rho\sin\theta\sin\varphi$
$z(\rho,\theta,\varphi)=c\rho\cos\varphi$

The Jacobian for this change of coordinates is

$\mathbf J=\begin{bmatrix}a\cos\theta\sin\varphi&-a\rho\sin\theta\sin\varphi&a\rho\cos\theta\cos\varphi\\b\sin\theta\sin\varphi&b\rho\cos\theta\sin\varphi&b\rho\sin\theta\cos\varphi\\c\cos\varphi&0&-c\rho\sin\varphi\end{bmatrix}$

for which $|\det\mathbf J|=abc\rho^2\sin\varphi$ .

Denoting the space bounded by the ellipsoid by $\mathcal V$ , the volume is given by the volume integral

$\displaystyle\iiint_{\mathcal V}\mathrm dV=abc\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=0}^{\rho=1}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{4abc\pi}3$

0,0(0 оценок)

Find the volume of the ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1

Ответ:

Ответ:

Ask your question to the AI advisor

More tips

Answers on questions: Mathematics

Ask an AI advisor a question