Using spherical coordinates, evaluate ∫∫∫ ye−(x^2+y^2+z^2)2 dV , V
where V is the solid that lies between y = 0 and hemisphere x^2 +y^2 +z^2 = 1 in the (right) half space y > 0.
How to solve this problem using spherical coords?
2014-09-23 11:49 am
回答 (1)
2014-09-23 5:24 pm
Since we want y > 0, we take θ in [-π/2, π/2].
So, ∫∫∫ ye^(-(x²+y²+z²)²) dV
= ∫(θ = -π/2 to π/2) ∫(φ = 0 to π) ∫(ρ = 0 to 1) (ρ sin θ sin φ)e^(-(ρ²)²) (ρ² sin φ dρ dφ dθ), via spherical coordinates
= ∫(θ = -π/2 to π/2) sin θ dθ * ∫(φ = 0 to π) sin² φ dφ *
∫(ρ = 0 to 1) ρ³ e^(-ρ⁴) dρ
= 0, due to the first factor (its integrand is odd).
I hope this helps!
So, ∫∫∫ ye^(-(x²+y²+z²)²) dV
= ∫(θ = -π/2 to π/2) ∫(φ = 0 to π) ∫(ρ = 0 to 1) (ρ sin θ sin φ)e^(-(ρ²)²) (ρ² sin φ dρ dφ dθ), via spherical coordinates
= ∫(θ = -π/2 to π/2) sin θ dθ * ∫(φ = 0 to π) sin² φ dφ *
∫(ρ = 0 to 1) ρ³ e^(-ρ⁴) dρ
= 0, due to the first factor (its integrand is odd).
I hope this helps!
收錄日期: 2021-04-21 00:31:17
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