Evaluate the surface integral ∫∫ (3x i + 2y j). dS, where S is the sphere given by x2 + y2 + z2 = 9.

To evaluate the surface integral ∫∫ (3x i + 2y j) · dS over the sphere defined by x² + y² + z² = 9, we apply the parameterization of the sphere and use the surface element.

The standard approach involves using the divergence theorem, but in this case, we will directly compute the surface integral.

1. Recognize that the surface S is a sphere of radius 3.
2. The unit normal to the surface of the sphere is a radial vector pointing outward from the center, given by n = (x/3, y/3, z/3) on the surface of the sphere.
3. The differential area element dS on the surface of the sphere is equal to dS = R² sin(θ) dθ dφ, where R = 3.
4. The surface integral can be computed as:

∫∫_S (3x i + 2y j) · dS = ∫∫_S (3x i + 2y j) · (n dS).

5. In spherical coordinates, let:

– x = 3 sin(θ) cos(φ)

– y = 3 sin(θ) sin(φ)

– z = 3 cos(θ)

where θ ranges from 0 to π and φ ranges from 0 to 2π.

The integral