Evaluate the triple integral ∭ExdV
where E is the solid bounded by the paraboloid x=4y^2+4z^2 and x=4
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To evaluate the triple integral ∭ExdV, we need to determine the limits of integration and the integrand.
The solid E is bounded by the paraboloid x = 4y^2 + 4z^2 and the plane x = 4. From these equations, we can see that the paraboloid extends from the plane x = 4 to the curve x = 4y^2 + 4z^2.
To determine the limits of integration, we can express the paraboloid in terms of either y or z. Let's solve for z:
x = 4y^2 + 4z^2
4z^2 = x - 4y^2
z^2 = (x - 4y^2) / 4
z = ±√((x - 4y^2) / 4)
Now, let's express the limits of integration for each variable:
For z, the limits range from -√((x - 4y^2) / 4) to √((x - 4y^2) / 4). This takes into account both the positive and negative square root solutions.
For y, the limits range from -√(x/4) to √(x/4). This can be derived by setting x = 4y^2 + 4z^2 to determine the range of y values.
For x, the limits range from 0 to 4, as the paraboloid is bounded by the plane x = 4.
Therefore, the triple integral becomes:
∭ExdV = ∫(x = 0 to 4) ∫(y = -√(x/4) to √(x/4)) ∫(z = -√((x - 4y^2) / 4) to √((x - 4y^2) / 4)) E(x, y, z) dzdydx
The integrand, E(x, y, z), depends on what is being integrated. Since it is not specified in the problem, we cannot determine the specific integrand without additional information.
To evaluate the triple integral ∭ExdV, we will use the cylindrical coordinate system since the given solid is described in terms of cylindrical coordinates.
In cylindrical coordinates, the equation of the paraboloid is:
x = 4y^2 + 4z^2
Since the solid is bounded by the paraboloid and the plane x=4, we can determine the limits of integration as follows:
1. For the z-limits: The paraboloid is symmetric with respect to the xy-plane, so we can integrate from z=0 to z=h, where h is the value of z that satisfies x=4. Substituting x=4 into the equation of the paraboloid gives:
4 = 4y^2 + 4z^2
This simplifies to:
y^2 + z^2 = 1
which represents a cylinder of radius 1 centered at the z-axis. Therefore, the limits for z are z=0 to z=1.
2. For the θ-limits: The solid is not dependent on θ, so we can integrate over the entire range of θ, which is from θ=0 to θ=2π.
3. For the r-limits: To determine the r-limits, we need to consider the projection of the solid onto the xy-plane. In this case, the projection is a circle centered at the origin with radius r=2. Therefore, the limits for r are r=0 to r=2.
Now, we can set up the triple integral:
∭ExdV = ∫(θ=0 to 2π) ∫(r=0 to 2) ∫(z=0 to 1) r Ex dz dr dθ
Since we are not given a specific function Ex(x,y,z) to integrate, we cannot provide a numerical value for the triple integral. However, the setup shown above gives the general form to evaluate the triple integral for any given function Ex(x,y,z).