Evaluate the triple integral ∭ExdV

Question

where E is the solid bounded by the paraboloid x=4y^2+4z^2 and x=4
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Answer 1

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.