What math functions are supported in expressions?

You can use standard functions: sin, cos, tan, asin, acos, atan, sqrt, abs, exp, log (natural log), log2, log10, pow(a,b), and the constant PI. Use * for multiplication.

How accurate is the result?

Accuracy depends on the quadrature order. Order 10 gives results accurate to 4–6 significant figures for smooth functions. Increase order for functions with rapid variation. Discontinuous or highly oscillatory functions may need special treatment beyond this tool.

How do I compute a plain surface area?

Set f(x, y, z) = 1. The integral then reduces to the surface area element integrated over the domain, giving the total surface area.

What is the surface area element used?

The tool computes the cross product of the partial derivatives ∂r/∂u and ∂r/∂v numerically using finite differences, then takes the magnitude. This |∂r/∂u × ∂r/∂v| is the surface area element multiplied by f at each quadrature point.

Can I integrate over a sphere or cylinder?

Yes. For a sphere of radius R: x = R*sin(u)*cos(v), y = R*sin(u)*sin(v), z = R*cos(u), with u in [0, PI] and v in [0, 2*PI]. For a cylinder of radius R and height H: x = R*cos(u), y = R*sin(u), z = v, with u in [0, 2*PI] and v in [0, H].

Surface Integral Calculator

Numerically evaluate surface integrals of scalar functions over parametric surfaces using 2D Gaussian quadrature.

Scalar Function f(x, y, z)

Use: x, y, z, sin, cos, sqrt, exp, log, PI, pow(a,b)

Parametric Surface r(u, v)

x =

y =

z =

Parameter Domain

u min

u max

v min

v max

Quadrature Order

Points per axis (3–20). Higher = more accurate but slower.

Quick Presets

Result

Fill in the form and click Compute.

Sample Grid (first 25 points)

Shows f(x,y,z) · |dS| at sampled quadrature nodes

u	v	x	y	z	f	\|dS\|

Summary

This tool numerically evaluates surface integrals of scalar functions f(x, y, z) over a parametric surface defined by x(u,v), y(u,v), z(u,v). It uses 2D Gaussian quadrature to approximate the integral of f times the surface area element |∂r/∂u × ∂r/∂v| over the parameter domain [u_min, u_max] × [v_min, v_max]. Results are shown with step-by-step breakdowns of the cross product magnitude, integrand sampling, and weighted sums.

How it works

Define your scalar function f(x, y, z) using standard math syntax (e.g. x*x + y*y).
Define the parametric surface: enter x(u,v), y(u,v), z(u,v) expressions.
Set the parameter domain: u range [u_min, u_max] and v range [v_min, v_max].
Choose quadrature order (5–15 points per axis) for accuracy.
Click Compute to evaluate the integral and view the step-by-step breakdown.

Use cases

Compute the surface area of a parametric surface (set f = 1).
Evaluate flux-related integrals in physics and engineering.
Verify hand-calculated surface integrals for calculus coursework.
Explore how integrand values vary across a surface using the sample grid.
Compute the mass of a thin shell with variable density f(x,y,z).
Integrate temperature or pressure distributions over curved surfaces.

Frequently Asked Questions

Last updated: 2026-06-18 · Reviewed by Nham Vu