SurfaceIntegrate

SurfaceIntegrate[f, {x, y, ...} ∈ surface] computes the scalar surface integral of the function f[x, y, ...] over the surface.

• SurfaceIntegrate[{p, q, ...}, {x, y, ...} ∈ surface] computes the vector surface integral of the vector field {p[x, y, ...], q[x, y, ...], ...}.

Examples

Surface area of a sphere:

SurfaceIntegrate[1, {x, y, z} ∈ Sphere[]]
(* 4 π *)

Flux through a hemisphere:

SurfaceIntegrate[{x, y, z}, {x, y, z} ∈ 
  ImplicitRegion[x^2 + y^2 + z^2 == 1 && z >= 0, {x, y, z}]]

Integrate a scalar field over a surface:

SurfaceIntegrate[x^2, {x, y, z} ∈ Sphere[{0, 0, 0}, 2]]

Please visit the official Wolfram Language Reference for more details.