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Surface Integral Calculator

Compute a surface integral ∬ f dS over a projected region.

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∬_S f dS = ∬ f·√(1+f_x²+f_y²) dxdy
integrand of z; use 1 for area
e.g. x^2+y^2

📖 Tutorial | Surface Integral

1. Definition
The scalar surface integral ∬_S f dS uses dS=√(1+φ_x²+φ_y²)dxdy. With f=1 it is the surface area.
2. Symbols
z=φ(x,y)surface
dSsurface element
√(1+φ_x²+φ_y²)area factor
3. How it works
  • Evaluate height and partials;
  • Multiply by the area factor;
  • 2D Simpson integration.
4. Steps
  1. Enter f (default 1 for area);
  2. Enter the surface and projection;
  3. Click Calculate.
5. Example
Example: area of z=x²+y² over [−1,1]².
Solution: dS=√(1+4x²+4y²)dxdy ≈ 7.45.
6. Pitfalls
This uses z=φ(x,y) projection;
f may use z;
f=1 gives area.

❓ FAQ | Surface Integral

Type 1 vs Type 2?
Type 1 over area, orientation-free; Type 2 is flux.
dS from?
Cross product of tangent vectors.
Surface area?
Set f=1.
Implicit surface?
Make it explicit z=φ(x,y).
Small difference?
Grid resolution.