[Solution] Practice: Triple integration (spherical)

Practice: Triple integration (spherical)
Write down an integral using spherical coordinates representing ∭E(x2+y2)   ⁣dV\displaystyle\iiint_E(x^2+y^2) \ \de{V}∭E​(x2+y2)  dV where EEE is the part of the positive octant inside the sphere x2+y2+z2=4x^2+y^2+z^2 = 4x2+y2+z2=4, outside the sphere x2+y2+z2=2x^2+y^2+z^2 = 2x2+y2+z2=2, and inside the cone 2z=x2+y22z =\sqrt{x^2 + y^2}2z=x2+y2​.

∫0π/2∫0arctan⁡2∫22ρ4sin⁡3ϕ dρ dϕ dθ\displaystyle\int^{\pi/2}_0\int^{\arctan 2}_0\int^2_{\sqrt{2}}\rho^4\sin^3\phi\ d\rho\ d\phi\ d\theta∫0π/2​∫0arctan2​∫2​2​ρ4sin3ϕ dρ dϕ dθ

∫0π∫0arctan⁡2∫22ρ4sin⁡3ϕ dρ dϕ dθ\displaystyle\int^{\pi}_0\int^{\arctan 2}_0\int^2_{\sqrt{2}}\rho^4\sin^3\phi\ d\rho\ d\phi\ d\theta∫0π​∫0arctan2​∫2​2​ρ4sin3ϕ dρ dϕ dθ

∫0π∫0arctan⁡2∫22ρ3sin⁡3ϕ dρ dϕ dθ\displaystyle\int^{\pi}_0\int^{\arctan 2}_0\int^2_{\sqrt{2}}\rho^3\sin^3\phi\ d\rho\ d\phi\ d\theta∫0π​∫0arctan2​∫2​2​ρ3sin3ϕ dρ dϕ dθ

∫0π/2∫0arctan⁡2∫22ρ3sin⁡3ϕ dρ dϕ dθ\displaystyle\int^{\pi/2}_0\int^{\arctan 2}_0\int^2_{\sqrt{2}}\rho^3\sin^3\phi\ d\rho\ d\phi\ d\theta∫0π/2​∫0arctan2​∫2​2​ρ3sin3ϕ dρ dϕ dθ

None of the above

I don't know

More Triple Integration (spherical) Questions:

Practice: Triple integration (spherical)

Practice: Triple integration (spherical)
Write down an integral using spherical coordinates representing ∭E(x2+y2)   ⁣dV\displaystyle\iiint_E(x^2+y^2) \ \de{V}∭E​(x2+y2)  dV where EEE is the part of the positive octant inside the sphere x2+y2+z2=4x^2+y^2+z^2 = 4x2+y2+z2=4, outside the sphere x2+y2+z2=2x^2+y^2+z^2 = 2x2+y2+z2=2, and inside the cone 2z=x2+y22z =\sqrt{x^2 + y^2}2z=x2+y2​.

Practice: Triple integration (spherical)

Practice: Triple integration (spherical)
Write down an integral using spherical coordinates representing ∭E(x2+y2)   ⁣dV\displaystyle\iiint_E(x^2+y^2) \ \de{V}∭E​(x2+y2)  dV where EEE is the part of the positive octant inside the sphere x2+y2+z2=4x^2+y^2+z^2 = 4x2+y2+z2=4, outside the sphere x2+y2+z2=2x^2+y^2+z^2 = 2x2+y2+z2=2, and inside the cone 2z=x2+y22z =\sqrt{x^2 + y^2}2z=x2+y2​.

Practice: Triple integration (spherical)

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Triple Integration (spherical)

Practice: Triple integration (spherical)

More Triple Integration (spherical) Questions:

Practice: Triple integration (spherical)

Practice: Triple integration (spherical)