Let and S be the part of the sphere between the planes and , oriented outward. The integral is equal to which of the following?

Question

Let  and S be the part of the sphere  between the planes  and  , oriented outward. The integral  is equal to which of the following?

Quizplus · Accepted Answer

Using the divergence theorem, the given line integral can be converted to a surface integral. The divergence of the vector field is 4x+2y+6z, and the given surface S has the equation z = x^2 + y^2. Using spherical coordinates, we have:

- 0 <= r <= 2
- 0 <= theta <= 2*pi
- 0 <= phi <= pi/4

The unit normal vector to S is (2x, 2y, -1), so the outward orientation means we use the negative of the divergence. Putting it all together, we get:

∫(S) F · dS = ∫∫(R) (-4r^2 sin(phi)) dr dtheta dphi = -8/3 pi

Using the given options, we see that only option A matches this value in hexadecimal format.

Let and S Be the Part of the Sphere