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A line integral calculator (or line integral solver) helps you evaluate the line integral and calculate line integrals along curves. Use this free parametric integral calculator and vector integral calculator to compute work, circulation, and flux. This line integral calculator supports line integral formulas for scalar and vector fields. Learn how to calculate line integral with step-by-step solutions.
Last updated: February 2, 2026
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Choose the type of line integral
Parameter interval [a, b] for the curve parametrization
Example Path:
Curve: r(t) = (t, t²)
Vector field: F(x,y) = ⟨x, y⟩
∫_C F·dr =
3
Work done by field
Arc Length (∫_C ds):
1.414
Length of the curve
Work Done:
3
Energy transferred along path
Formulas:
Solution Steps:
Line Integral Concepts:
Formula
∫_C f ds
Mass, charge, or values along curve
Formula
∫_C F·dr
Work integral along path
Formula
∮_C F·dr
Rotation tendency around loop
Method
r(t) = ⟨x(t), y(t)⟩
Convert to parameter integral
Formula
∫_C ds = ∫_a^b |r'(t)| dt
Total curve length
Physics
W = ∫_C F·dr
Work by force along path
Vector field line integral along parametric curve
Line Integral
3.0
∫_C F·dr
Arc Length
1.414
∫_C ds
A line integral solver (or line integral calculator) evaluates line integrals along curves in vector and scalar fields. To evaluate the line integral: 1) Parametrize the curve as r(t) = ⟨x(t), y(t)⟩ for t ∈ [a,b]. 2) Find r'(t) = ⟨x'(t), y'(t)⟩. 3) For scalar fields: calculate ∫_a^b f(r(t))|r'(t)| dt. 4) For vector fields: calculate ∫_a^b F(r(t))·r'(t) dt. This line integral solver automates these steps.
A parametric integral calculator evaluates integrals along parametrized curves by converting line integrals to standard integrals using r(t) = ⟨x(t), y(t)⟩. A vector integral calculator (or vector line integral calculator) computes line integrals of vector fields ∫_C F·dr, calculating work done, circulation, and flux. This parametric integral calculator and vector integral calculator handle both scalar and vector field integrals.
Line integral formulas include: Scalar field: ∫_C f ds = ∫_a^b f(r(t))|r'(t)| dt. Vector field: ∫_C F·dr = ∫_a^b F(r(t))·r'(t) dt. Arc length: ds = |r'(t)| dt. A path integral calculator (or line integral calculator) evaluates integrals along paths using these line integral formulas. These formulas convert curve integrals to parameter integrals for evaluation.
Our line integral calculator evaluates integrals along parametric curves in vector and scalar fields. The calculator uses parametric integration techniques to convert line integrals into standard integrals over parameter intervals.
Scalar Field Line Integral:
∫_C f(x,y) ds = ∫_a^b f(r(t)) |r'(t)| dtWhere r(t) parametrizes curve C
Vector Field Line Integral (Work):
∫_C F·dr = ∫_a^b F(r(t))·r'(t) dtDot product of field and velocity
Arc Length Element:
ds = |r'(t)| dt = √[(dx/dt)² + (dy/dt)²] dtInfinitesimal arc length
Green's Theorem (Circulation):
∮_C P dx + Q dy = ∬_D (∂Q/∂x - ∂P/∂y) dAConverts line integral to area integral
Line integrals extend the concept of integration to curves in space. For scalar fields, the line integral represents the area of a "curtain" hanging from the curve in the field. For vector fields, it represents work done by a force field moving an object along the path, or flux through the curve. The key is parametrizing the curve and converting to a standard integral.
Need other vector calculus tools? Check out our partial derivative calculator and cross product calculator.
Get Custom Calculator for Your PlatformResult: ∫_C F·dr = 1
Work done by the vector field F along the parabolic path is 1 unit.
∮_C F·dr around circle of radius R
Use Green's theorem or direct parametrization
∫_C ∇f·dr = f(end) - f(start)
Path-independent (Fundamental Theorem)
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