Use an appropriate coordinate system to find the volume of the given solid.
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Use an appropriate coordinate system to find the volume of the given solid.

[From: ] [author: ] [Date: 12-04-23] [Hit: ]
z = √(x^2 + y^2) ==> ρ cos φ = ρ sin φ ==> φ = π/4.Projecting the region onto the xy-plane yields a portion of the circle x^2 + y^2 = 3 (from intersecting the surfaces given) between x = 0 and y = x ==> θ is in [π/4, π/2].So,= (π/3)(2 - √2).I hope this helps!......
The solid below (x^2)+(y^2)+(z^2)=4, above z=sqrt((x^2)+(y^2)), between y=x, and x=0. with y greater than or equal to 0.

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Use spherical coordinates.
x^2 + y^2 + z^2 = 4 ==> ρ = 2
z = √(x^2 + y^2) ==> ρ cos φ = ρ sin φ ==> φ = π/4.

Projecting the region onto the xy-plane yields a portion of the circle x^2 + y^2 = 3 (from intersecting the surfaces given) between x = 0 and y = x ==> θ is in [π/4, π/2].

So, the volume ∫∫∫ 1 dV equals
∫(θ = π/4 to π/2) ∫(φ = 0 to π/4) ∫(ρ = 0 to 2) 1 * (ρ^2 sin φ dρ dφ dθ)
= ∫(θ = π/4 to π/2) dθ * ∫(φ = 0 to π/4) sin φ dφ * ∫(ρ = 0 to 2) ρ^2 dρ
= (π/4) * (1 - √2/2) * (8/3)
= (π/3)(2 - √2).

I hope this helps!
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