Surface area: A = ∬D |rᵤ × rᵥ| dA where r(u,v) parametrizes the surface. For z = f(x,y): A = ∬ √(1+fₓ²+fᵧ²) dA. Sphere (r,θ,φ): A = 4πr². Cylinder: A = 2πrh. The cross product magnitude gives the infinitesimal area element.
How to Use
Select a surface:
- r(u,v): Parametric surface
- Domain: u,v ranges
- Output: Surface area
Formula Derivation
|rᵤ × rᵥ| = area of parallelogram spanned by tangent vectors. Summing infinitesimal parallelograms gives total surface area. For revolution surfaces: A = 2π∫ r(x)√(1+r'²) dx.
Classic Results
- Sphere: 4πr²
- Cylinder: 2πrh
- Cone: πr√(r²+h²)
- Torus: 4π²Rr
Step-by-Step Instructions
- 1Select surface.
- 2View parametrization.
- 3Compute cross product.
- 4Integrate magnitude.
- 5Get surface area.