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[tex]\begin{cases}x(\rho,\theta,\varphi)=\rho\cos\theta\sin\varphi\\y(\rho,\theta,\varphi)=\rho\sin\theta\sin\varphi\\z(\rho,\theta,\varphi)=\rho\cos\varphi\end{cases}\implies\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]

[tex]x^2+y^2=\rho^2\cos^2\theta\sin^2\varphi+\rho^2\sin^2\theta\sin^2\varphi=\rho^2\sin^2\varphi[/tex]

[tex]\displaystyle\iiint_E(x^2+y^2)\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=1}^{\rho=5}\rho^4\sin^3\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]
[tex]=\displaystyle2\pi\left(\int_{\varphi=0}^{\varphi=\pi}\sin^3\varphi\,\mathrm d\varphi\right)\left(\int_{\rho=1}^{\rho=5}\rho^4\,\mathrm d\rho\right)[/tex]
[tex]=\dfrac{24992\pi}{15}[/tex]

To solve this problem, it is necessary determine the integrations limits first.

The solution is:

V = 1666,13×π vol units

In order to express the information in spherical coordinates, we have:

x = ρ×cosθ×sinφ

y = ρ×sinθ×sinφ

z = ρ×cosφ

and     dV = ρ²×sinφ×dρ×dθ×dφ

0 ≤ φ ≤ π

0 ≤ θ ≤ 2×π

x² + y² + z² = 25       and         x² + y² + z² = 1    are two concentric spheres with center at the origin and radius 5 and 1 respectevely .

According to that:

1 ≤  ρ ≤ 5

V = ∫∫∫ ( x² + y² )× dV  

V  =   ∫∫∫ ( x² + y² )×ρ²×sinφ×dρ×dθ×dφ

V =  ∫∫∫ (  ρ²×cos²θ×sin²φ  + ρ²×sin²θ×sin²φ ) ×ρ²×sinφ×dρ×dθ×dφ

V =  ∫∫∫ [  ρ²×sin²φ ( cos²θ + sin²θ )  ×ρ²×sinφ×dρ×dθ×dφ

V =  ∫∫∫ [  ρ²×sin²φ×ρ²×sinφ×dρ×dθ×dφ

V =  ∫∫∫ [ ρ⁴×sin³φ×dρ×dθ×dφ

V =  ∫₁⁵ρ⁴×dρ ×∫dθ ×∫sin³φ×dφ

V =  ρ⁵/5 |₁⁵×∫dθ ×∫sin³φ×dφ

ρ⁵/5 |₁⁵  = 5⁵/5 - 1⁵/5   = 3125 / 5 - 1/5  = 3124/5

V = 3124/5× θ |₀²π×∫sin³φ×dφ

θ |₀²π  = 2×π - 0

V = 3124/5× (2×π) ×∫sin³φ×dφ

∫sin³φ×dφ  [ 0  y π ] = 4/3   (from Simbolab)

V = 3124/5× (2×π) ×4/3

V = 1666,13×π   vol units

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