Mathematics: Evaluate dx dy dz (x2 y2 z2)3/2 w , where w is the solid bounded by the two spheres x2 y2 z2

I dont think it works like that dude...

Evaluate dx dy dz (x2 y2 z2)3/2 w , where w is the - id222321110309, rogersdanny7347

Ответ:

76KamLan

13.03.2022

Looks like the integral is supposed to be

$\displaystyle\iiint_{\mathcal W}\frac{\mathrm dx\,\mathrm dy\,\mathrm dz}{(x^2+y^2+z^2)^{3/2}}$

but it's not clear what spheres bound the region $\mathcal W$ . I'll assume it's the space between two spheres, both centered at the origin, with radii

and

such that

.
Convert to spherical coordinates, setting

$\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\theta\end{cases}$

Then the region $\mathcal W$ is parameterized by varying over $r\le\rho\le R$ , $0\le\theta\le2\pi$ , and $0\le\varphi\le\pi$ . Note that $x^2+y^2+z^2=\rho^2$ . We also have the Jacobian

$\mathbf J=\dfrac{\partial(x,y,z)}{\partial(\rho,\theta,\varphi)}=\begin{bmatrix}\cos\theta\sin\varphi&-\rho\sin\theta\sin\varphi&\rho\cos\theta\cos\varphi\\\sin\theta\sin\varphi&\rho\cos\theta\sin\varphi&\rho\sin\theta\cos\varphi\\\cos\varphi&0&-\rho\sin\varphi\end{bmatrix}$

whose determinant is

$\det\mathbf J=-\rho^2\sin\varphi$

Under this change of coordinates, we take

$\mathrm dx\,\mathrm dy\,\mathrm dz=|\det\mathbf J|\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$

So, we have

$\displaystyle\iiint_{\mathcal W}\frac{\mathrm dx\,\mathrm dy\,\mathrm dz}{(x^2+y^2+z^2)^{3/2}}=\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=r}^{\rho=R}\frac{\rho^2\sin\varphi}{(\rho^2)^{3/2}}\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$
$=\displaystyle2\pi\int_{\varphi=0}^{\varphi=\pi}\int_{\rho=r}^{\rho=R}\frac{\sin\varphi}\rho\,\mathrm d\rho\,\mathrm d\varphi$
$=\displaystyle2\pi\left(\int_0^\pi\sin\varphi\,\mathrm d\varphi\right)\left(\int_r^R\frac{\mathrm d\rho}\rho\right)$
$=2\pi\cdot2\cdot(\ln|R|-\ln|r|)$
$=4\pi\ln\dfrac Rr$

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Evaluate dx dy dz (x2 y2 z2)3/2 w , where w is the solid bounded by the two spheres x2 y2 z2

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