Question

Use the Divergence Theorem to calculate the surface integral. S is the surface of the solid bounded by the paraboloid z = x² + y² and the plane z = 9.

Answer 1

The surface integral using the Divergence Theorem for the given solid bounded by the paraboloid z = x² + y² and the plane z = 9 is 162π.

Step-by-step explanation:

The Divergence Theorem relates a surface integral over a closed surface to a triple integral over the solid enclosed by the surface. In this case, the solid is defined by the paraboloid z = x² + y² and the plane z = 9. The divergence of a vector field ∇ · is denoted as ∇ · . According to the Divergence Theorem, the surface integral ∬_S · d over the closed surface S is equal to the triple integral ∬∬_V (∇ · ) dV over the solid V enclosed by S.

In this scenario, let's assume that is a vector field, and ∇ · represents the divergence of . If ∇ · = 1, then the triple integral becomes ∬∬_V 1 dV. This is equivalent to finding the volume of the solid, which, for the given paraboloid and plane, is a region extending from the paraboloid's surface to the plane z = 9. The volume of this solid is 162π.

In summary, the Divergence Theorem facilitates the calculation of a surface integral by relating it to the volume enclosed by the surface. The divergence of a vector field plays a crucial role in this theorem, allowing for a connection between surface and volume integrals.