141k views
1 vote
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.

1 Answer

3 votes

Final Answer:

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.

User Aseem Kishore
by
8.5k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories