Question

Evaluate the triple integral ∭6x(x^2 + y^2) dv, where E is the solid in the first octant that lies beneath the paraboloid z = 1 − x^2 − y^2.

a. 0
b. 1
c. 2
d. 3

Answer 1

To evaluate the given triple integral, we need to determine the limits of integration for each variable x, y, and z, based on the equation of the paraboloid and the region defined as E. Once we have the limits set up, we can compute the integral to find the answer.

Step-by-step explanation:

We are given the triple integral ∭6x(x^2 + y^2) dv, with the solid region E defined as the region in the first octant beneath the paraboloid z = 1 − x^2 − y^2.

To evaluate this triple integral, we need to determine the limits of integration for each variable x, y, and z. In this case, since E is defined as the region beneath the paraboloid, the limits for x, y, and z are all dependent on the equation of the paraboloid.

By setting up and evaluating these limits properly, we can then compute the integral to find the answer.