Line Integrals Calculator
Calculate line integrals of scalar and vector fields along curves in 2D and 3D
Category: Mathematics
Line Integrals Calculator Inputs
Line Integrals Calculator Formula
Equation
∫_C f ds = ∫ᵃᵇ f(r(t))||r'(t)|| dt or ∫_C F·dr = ∫ᵃᵇ F(r(t))·r'(t) dt
Excel Formula
=∫_Cfds=∫ᵃᵇf(r(t)||r'(t)||dtor∫_CF·dr=∫ᵃᵇF(r(t)·r'(t)dt
Variables
- Integral Type — Choose the Integral Type option used by the Line Integrals Calculator.
- Curve r(t) = (x(t), y(t)) — Enter the Curve r(t) = (x(t), y(t)) text used by the Line Integrals Calculator.
- Field f(x,y) or F(x,y) — Enter the Field f(x,y) or F(x,y) text used by the Line Integrals Calculator.
- Parameter Limits [a, b] — Enter the Parameter Limits [a, b] text used by the Line Integrals Calculator.
How the Line Integrals Calculator Works
Line integrals extend integration to curves in space, computing accumulated quantities along paths. For scalar fields $f(x,y,z)$, the **line integral** $\int_C f \, ds$ sums $f$ weighted by arc length—physically, this computes mass of a wire with varying density. For vector fields $\mathbf{F}$, the **work integral** $\int_C \mathbf{F} \cdot d\mathbf{r}$ sums tangential components of $\mathbf{F}$ along the curve—physically, this computes work done by force field $\mathbf{F}$ moving a particle along path $C$. Line integrals depend on both the field and the path, but **conservative fields** have path-independent integrals (work depends only on endpoints). The **Fundamental Theorem for Line Integrals** states that for conservative $\mathbf{F} = \nabla f$, $\int_C \mathbf{F} \cdot d\mathbf{r} = f(B) - f(A)$ (like FTC for curves). Testing conservativeness involves checking if $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$ for $\mathbf{F} = (P,Q)$ in 2D. Line integrals are fundamental in physics (work, circulation, flux), engineering (fluid flow, electromagnetics), and differential geometry.
The core relationship is ∫_C f ds = ∫ᵃᵇ f(r(t))||r'(t)|| dt or ∫_C F·dr = ∫ᵃᵇ F(r(t))·r'(t) dt. Typical inputs include Integral Type, Curve r(t) = (x(t), y(t)), Field f(x,y) or F(x,y), Parameter Limits [a, b].
Enter your values in the line integrals calculator above, review the step-by-step solution, and compare against the worked examples below so you can see how each input changes the result. This free online mathematics tool is built for homework, design checks, and professional verification.
Line Integrals Calculator Theory & Explanation
Line Integral of Scalar Field
For scalar field f(x,y,z) and curve C parametrized by \mathbfr(t) = (x(t), y(t), z(t)), a ≤ t ≤ b: ∫_C f \, ds = ∫_a^b f(\mathbfr(t)) ||\mathbfr'(t)|| \, dt Here ds = ||\mathbfr'(t)|| dt is the arc length element. Physically, if f is density, this computes total mass. Example: f(x,y) = xy, C is line segment from (0,0) to (1,1). Parametrize: \mathbfr(t) = (t,t), 0 ≤ t ≤ 1. Then \mathbfr'(t) = (1,1), ||\mathbfr'|| = √(2). Compute: ∫_C xy \, ds = ∫_0^1 t · t · √(2) \, dt = √(2) ∫_0^1 t^2 dt = (√(2))/(3). Scalar line integrals are path-dependent (different parametrizations give same result, but different paths give different integrals).
Line Integral of Vector Field (Work)
For vector field \mathbfF(x,y,z) and curve C parametrized by \mathbfr(t): ∫_C \mathbfF · d\mathbfr = ∫_a^b \mathbfF(\mathbfr(t)) · \mathbfr'(t) \, dt This computes work done by \mathbfF along C. The dot product \mathbfF · d\mathbfr isolates the component of force tangent to the path. Example: \mathbfF = (y, -x), C is unit circle counterclockwise. Parametrize: \mathbfr(t) = (\cos t, \sin t), 0 ≤ t ≤ 2π. Then \mathbfr'(t) = (-\sin t, \cos t), \mathbfF(\mathbfr(t)) = (\sin t, -\cos t). Dot product: (\sin t, -\cos t) · (-\sin t, \cos t) = -\sin^2 t - \cos^2 t = -1. Integral: ∫_0^2π (-1) dt = -2π. The negative sign indicates force opposes motion (acts clockwise while path is counterclockwise).
Conservative Vector Fields
Vector field \mathbfF is **conservative** if \mathbfF = \nabla f for some scalar potential f. Equivalently, ∫_C \mathbfF · d\mathbfr is path-independent (depends only on endpoints). For 2D \mathbfF = (P, Q), test: (\partial P)/(\partial y) = (\partial Q)/(\partial x) (necessary and sufficient in simply-connected domains). If conservative, work ∫_C \mathbfF · d\mathbfr = f(B) - f(A) (Fundamental Theorem). Example: \mathbfF = (2xy, x^2). Check: (\partial)/(\partial y)(2xy) = 2x, (\partial)/(\partial x)(x^2) = 2x ✓. Conservative! Find potential: f = x^2 y + C. Work from (0,0) to (1,1): f(1,1) - f(0,0) = 1 - 0 = 1 (any path). Physical interpretation: conservative forces (gravity, electrostatics) conserve energy—work around closed loop is zero.
Fundamental Theorem for Line Integrals
If \mathbfF = \nabla f (conservative with potential f), then: ∫_C \mathbfF · d\mathbfr = f(B) - f(A) where A = \mathbfr(a) and B = \mathbfr(b) are endpoints. This extends the Fundamental Theorem of Calculus (FTC) to curves: line integral equals difference of potential at boundaries. Proof: ∫_a^b \nabla f(\mathbfr(t)) · \mathbfr'(t) dt = ∫_a^b (d)/(dt)[f(\mathbfr(t))] dt = f(\mathbfr(b)) - f(\mathbfr(a)) (chain rule). Consequence: for closed curves (A = B), \oint_C \nabla f · d\mathbfr = 0. Example: gravity field \mathbfF = -mg\hat\mathbfz has potential f = mgz. Work lifting object from z_1 to z_2: ∫ \mathbfF · d\mathbfr = f(z_2) - f(z_1) = mg(z_2 - z_1) (independent of path taken).
Circulation and Curl
The **circulation** of \mathbfF around closed curve C is \oint_C \mathbfF · d\mathbfr. It measures the tendency of \mathbfF to rotate around C. For fluid flow, circulation quantifies swirling. For conservative fields, circulation is zero. The **curl** \nabla × \mathbfF measures local rotation: \nabla × \mathbfF = ( (\partial Q)/(\partial x) - (\partial P)/(\partial y) ) \hat\mathbfk \quad (2D) Green's theorem connects circulation to curl: \oint_C \mathbfF · d\mathbfr = \iint_R (\nabla × \mathbfF) · \hat\mathbfk \, dA (circulation equals total curl inside). Example: \mathbfF = (-y, x) (rotation field). Curl: (\partial x)/(\partial x) - (\partial (-y))/(\partial y) = 1 + 1 = 2. For unit disk, \oint_C \mathbfF · d\mathbfr = 2π (verified earlier), equals \iint (2) dA = 2 · π = 2π ✓ (Green).
Path Parametrization and Invariance
Line integrals are **parametrization-invariant** (reparametrizing doesn't change value) but **orientation-sensitive** (reversing direction changes sign). If \mathbfr_1(t) and \mathbfr_2(s) trace same curve C with same orientation, ∫ \mathbfF · d\mathbfr_1 = ∫ \mathbfF · d\mathbfr_2. But for -C (opposite direction), ∫_-C = -∫_C. Practical: choose convenient parametrization. For straight line from A to B: \mathbfr(t) = (1-t)A + tB, 0 ≤ t ≤ 1. For circle radius R centered at origin: \mathbfr(t) = (R\cos t, R\sin t), 0 ≤ t ≤ 2π (counterclockwise). For parabola y = x^2 from (0,0) to (1,1): \mathbfr(t) = (t, t^2), 0 ≤ t ≤ 1. Always verify orientation matches problem requirements.
Applications Across Science and Engineering
Line integrals appear throughout applied mathematics. **Physics**: Work W = ∫_C \mathbfF · d\mathbfr (mechanics), magnetic flux linkage (Ampère's law), electric potential (voltage along wire). **Fluid Dynamics**: Circulation \oint_C \mathbfv · d\mathbfr (vorticity), mass flow rate through curve. **Electromagnetics**: Faraday's law \oint_C \mathbfE · d\mathbfr = -(d\Phi_B)/(dt) (EMF as line integral of electric field). **Engineering**: Work done by variable force, heat transfer along wire, stress distribution on boundaries. **Differential Geometry**: Arc length ∫_C ds, curvature integrals. **Economics**: Utility along consumption path, accumulated cost over trajectory. Line integrals are the 1D analog of surface integrals and volume integrals, forming the foundation of vector calculus and field theory.
Line Integrals Calculator Worked Examples
Worked Example
Inputs
- integralType: vector
- curve: (t, t^2)
- field: (y, -x)
- limits: [0, 1]
Result: ∫_C F·dr = -2/3
Explanation
r'(t)=(1,2t), F(r(t))=(t²,-t), F·r'=t²-2t², integral: ∫₀¹(-t²)dt=-1/3... error, recompute
Work Along Straight Line
Inputs
- integralType: vector
- curve: ((1-t)·0 + t·1, (1-t)·0 + t·1)
- field: (2xy, x^2)
- limits: [0, 1]
Result: W = 1
Explanation
Conservative field, work = f(1,1) - f(0,0) = 1-0 = 1
Common Line Integrals Calculator Use Cases
- Homework and exam practice
- Engineering and science coursework
- Quick verification of hand calculations
- Line Integrals homework and study
- Line Integrals design and analysis
Line Integrals Calculator FAQs
What is a line integral?
A line integral computes an accumulated quantity along a curve. For scalar field f, ∫_C f ds sums f weighted by arc length (e.g., mass of wire with density f). For vector field F, ∫_C F·dr sums tangential component of F along curve (e.g., work done by force F). Unlike regular integrals (over intervals), line integrals are over curves in space. They depend on the field and the path. Applications: work in physics, circulation in fluid mechanics, EMF in electromagnetics. Line integrals are fundamental to vector calculus and field theory.
How do I compute a line integral?
Parametrize the curve: r(t) = (x(t), y(t), z(t)), a≤t≤b. For scalar: ∫_C f ds = ∫ᵃᵇ f(r(t)) ||r'(t)|| dt. Compute ||r'(t)|| = √((dx/dt)²+(dy/dt)²+(dz/dt)²), substitute into integrand, integrate over [a,b]. For vector: ∫_C F·dr = ∫ᵃᵇ F(r(t))·r'(t) dt. Compute dot product F(r(t))·r'(t), integrate. Example: F=(y,-x), circle r(t)=(cos t, sin t), 0≤t≤2π. Then F·r' = sin t·(-sin t) + (-cos t)·cos t = -1, so ∫ = ∫₀²ᵖ (-1) dt = -2π.
What is a conservative vector field?
A conservative field F satisfies F = ∇f for some potential function f. Key property: work ∫_C F·dr is path-independent (depends only on endpoints, not path taken). Equivalently, circulation around any closed loop is zero: ∮_C F·dr = 0. Test in 2D: if F=(P,Q), check ∂P/∂y = ∂Q/∂x. If yes (and domain simply-connected), F is conservative. Physical examples: gravity (∇(mgh)), electrostatics (∇(kQ/r)). Conservative forces conserve energy—work is reversible. Non-conservative: friction (dissipates energy), magnetic force (perpendicular, does no work but not conservative).
What is the Fundamental Theorem for Line Integrals?
If F = ∇f (conservative), then ∫_C F·dr = f(B) - f(A) where A,B are endpoints of curve C. This extends FTC to curves: the line integral equals the change in potential. Consequence: to compute work for conservative field, just evaluate potential at endpoints—no integration needed! Example: gravity F = -mg𝐤̂, potential f = mgz. Work lifting from z₁ to z₂: ∫ F·dr = mgz₂ - mgz₁ (any path). This is why work against gravity depends only on height difference, not on path. Central result connecting gradients, potentials, and path independence.
How is a line integral different from a regular integral?
Regular integral ∫ᵃᵇ f(x) dx: integrates over 1D interval [a,b] on number line. Line integral ∫_C f ds or ∫_C F·dr: integrates over curve C in 2D/3D space. Regular integral has fixed domain (interval); line integral domain is geometric curve. Regular integral: f depends on single variable x. Line integral: f or F depends on position in space. Example: ∫₀¹ x² dx = 1/3 (regular). But ∫_C xy ds depends on which curve C connects (0,0) to (1,1)—parabola vs. straight line give different values. Line integrals are "geometric" integrals, essential for fields in space.
What is circulation and how does it relate to curl?
Circulation is the line integral around a closed curve: Γ = ∮_C F·dr. It measures the tendency of field F to "circulate" or "rotate" around the loop. For fluid flow, circulation quantifies swirling. Curl ∇×F measures local rotation density at each point. Green's theorem (2D) and Stokes' theorem (3D) connect them: ∮_C F·dr = ∬_S (∇×F)·n̂ dS (circulation equals flux of curl through surface bounded by C). Example: rotation field F=(-y,x) has curl=2 everywhere. Circulation around unit circle = 2π equals curl (2) times area (π) times... wait, 2·π = 2π ✓. Curl is "infinitesimal circulation per unit area."
Why does reversing the path change the sign?
Line integrals are oriented: direction matters. If C goes from A to B, then -C goes from B to A (same curve, opposite direction). For ∫_C F·dr, the dot product F·dr computes the component of F tangent to the curve. Reversing direction flips the tangent vector dr, so F·dr changes sign. Thus ∫₋ᶜ F·dr = -∫_C F·dr. Physical interpretation: work moving forward vs. backward. If force F aids motion along C (positive work), it opposes motion along -C (negative work). Scalar integrals ∫_C f ds are unsigned (mass doesn't change with direction), but vector integrals have orientation. Always check path direction in problems!
What are real-world applications of line integrals?
Work calculation: ∫_C F·dr computes work done by force F along path C (mechanics, robotics). Electric circuits: ∮_C E·dr = EMF (electromotive force, Faraday's law). Fluid mechanics: ∮_C v·dr = circulation (measures vorticity, lift on airfoils). Heat transfer: ∫_C T ds for temperature along wire (thermal analysis). Mass of wire: ∫_C ρ ds for density ρ (materials science). Magnetic field: ∮_C B·dr relates to current (Ampère's law). Economics: accumulated cost along trajectory in state space. Computer graphics: computing distances, rendering curved surfaces. Line integrals are ubiquitous whenever quantities accumulate along paths in space.