Home

Line Integrals (Scalar) Calculator

Compute a scalar line integral ∫ f ds along a parametric curve.

展开更多 ▾
∫_C f ds = ∫ f(x(t),y(t))√(x'²+y'²) dt
e.g. x+y
e.g. t
e.g. t^2

📖 Tutorial | Line Integral (Scalar)

1. Definition
The scalar line integral ∫_C f ds = ∫ f(x(t),y(t))√(x'²+y'²) dt is independent of orientation.
2. Symbols
fline density
x(t),y(t)parametric curve
dsarc-length element
√(x'²+y'²)speed
3. How it works
  • Evaluate curve points and speed;
  • Multiply by speed to get ds;
  • Integrate by Simpson over t.
4. Steps
  1. Enter f, the curve and t range;
  2. Click Calculate.
5. Example
Example: f=x+y along x=t,y=t², t:0→1.
Solution: ∫₀¹(t+t²)√(1+4t²)dt ≈ 1.455.
6. Pitfalls
Orientation independent;
ds≥0; lower bound smaller;
Includes the speed factor.

❓ FAQ | Line Integral (Scalar)

Type 1 vs Type 2?
Type 1 uses arc length (orientation-free); Type 2 uses coordinates.
Why the speed factor?
ds=√(dx²+dy²).
Mass?
With f as density, yes.
Closed curve?
Parametrize around it.
Approximate?
Numerical, usually accurate.