To solve this triple integral, we will use cylindrical coordinates.
In cylindrical coordinates, the equation of the paraboloid x = 4y^2 + 4z^2 can be written as:
ρ^2 = 4y^2 + 4z^2
Simplifying this equation, we get:
ρ^2 = 4(y^2 + z^2)
This is the equation of a cylinder aligned with the z-axis and centered at the origin, with a radius of √4 = 2.
The equation x = 4 represents a plane that intersects the cylinder at x = 4, forming a circular cross-section.
To evaluate the triple integral, we need to set up the limits of integration.
In cylindrical coordinates, the limits for ρ are from 0 to 2, the limits for θ are from 0 to 2π, and the limits for z are from 0 to the value that satisfies the equation of the plane x = 4:
ρ: 0 ≤ ρ ≤ 2
θ: 0 ≤ θ ≤ 2π
z: 0 ≤ z ≤ (4 - 4y^2)/4
Now, let's calculate the triple integral:
∭ExdV = ∫(0 to 2) ∫(0 to 2π) ∫(0 to (4 - 4y^2)/4) ρ dρ dθ dz
Integrating with respect to ρ, we get:
= ∫(0 to 2) ∫(0 to 2π) [1/2ρ^2] (0 to (4 - 4y^2)/4) dθ dz
= ∫(0 to 2) ∫(0 to 2π) (1/2)(4 - 4y^2)/4 dθ dz
= ∫(0 to 2) ∫(0 to 2π) (1/2)(1 - y^2) dθ dz
Integrating with respect to θ, we get:
= ∫(0 to 2) [θ - (θy^2/2)] (0 to 2π) dz
= ∫(0 to 2) [2π - πy^2] dz
= [2πz - πy^2z] (0 to 2)
= [4π - 2πy^2] - 0
= 4π - 2πy^2
Therefore, the triple integral ∭ExdV = 4π - 2πy^2.
Evaluate the triple integral ∭ExdV
where E is the solid bounded by the paraboloid x=4y2+4z2 and x=4
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