Sign in to save

Bookmark this page so you can find it later.

Sign in to save

Bookmark this page so you can find it later.

Line Integrals & Surface Integrals Reference cheat sheet - grade college

Click image to open full size

Calculus Grade college

Line Integrals & Surface Integrals Reference Cheat Sheet

A printable reference covering scalar line integrals, vector line integrals, surface integrals, flux, Green’s theorem, Stokes’ theorem, and divergence theorem for college calculus.

TopicsIntegralsVectorsCalculusDerivativesFunctionsStudy Guide
Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

Line integrals and surface integrals extend single-variable and multivariable integration to curves and surfaces. This reference helps students choose the correct integral form, parameterize the geometry, and connect computations to physical meaning. These tools are essential in vector calculus, especially for work, circulation, flux, mass, and field flow.

A compact cheat sheet is useful because many errors come from mixing up dsds, drd\mathbf{r}, and dSdS.

The core idea is to rewrite a curve or surface using parameters, then convert the integral into an ordinary single or double integral. Scalar line integrals use ds=∥r′(t)∥ dtds = \|\mathbf{r}'(t)\|\,dt, while vector line integrals use dr=r′(t) dtd\mathbf{r} = \mathbf{r}'(t)\,dt. Surface integrals use the area element dSdS or the oriented vector area element ru×rv dA\mathbf{r}_u \times \mathbf{r}_v\,dA.

The major theorems, including Green’s theorem, Stokes’ theorem, and the divergence theorem, convert difficult integrals into simpler boundary or region integrals when their hypotheses are met.

Key Facts

  • For a parametrized curve r(t)\mathbf{r}(t) on a≤t≤ba \le t \le b, the scalar line integral is ∫Cf ds=∫abf(r(t))∥r′(t)∥ dt\int_C f\,ds = \int_a^b f(\mathbf{r}(t))\|\mathbf{r}'(t)\|\,dt.
  • For a vector field F\mathbf{F} along a curve CC, the work integral is ∫CF⋅dr=∫abF(r(t))⋅r′(t) dt\int_C \mathbf{F}\cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot \mathbf{r}'(t)\,dt.
  • Reversing the orientation of a vector line integral changes the sign, so ∫−CF⋅dr=−∫CF⋅dr\int_{-C} \mathbf{F}\cdot d\mathbf{r} = -\int_C \mathbf{F}\cdot d\mathbf{r}.
  • A scalar line integral does not change sign when orientation is reversed because ds=∥r′(t)∥ dtds = \|\mathbf{r}'(t)\|\,dt is always nonnegative.
  • For a parametrized surface r(u,v)\mathbf{r}(u,v) over a parameter region DD, the scalar surface integral is ∬Sf dS=∬Df(r(u,v))∥ru×rv∥ dA\iint_S f\,dS = \iint_D f(\mathbf{r}(u,v))\|\mathbf{r}_u \times \mathbf{r}_v\|\,dA.
  • The flux of F\mathbf{F} through an oriented surface SS is ∬SF⋅n dS=∬DF(r(u,v))⋅(ru×rv) dA\iint_S \mathbf{F}\cdot \mathbf{n}\,dS = \iint_D \mathbf{F}(\mathbf{r}(u,v))\cdot(\mathbf{r}_u \times \mathbf{r}_v)\,dA.
  • Green’s theorem states that ∮CP dx+Q dy=∬R(∂Q∂x−∂P∂y)dA\oint_C P\,dx + Q\,dy = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA for a positively oriented simple closed curve CC bounding RR.
  • Stokes’ theorem states that ∮CF⋅dr=∬S(∇×F)⋅n dS\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F})\cdot \mathbf{n}\,dS, and the divergence theorem states that ∬SF⋅n dS=∭E∇⋅F dV\iint_S \mathbf{F}\cdot \mathbf{n}\,dS = \iiint_E \nabla\cdot\mathbf{F}\,dV for a closed surface SS.

Vocabulary

Scalar line integral
An integral of a scalar function along a curve that adds weighted arc length using ∫Cf ds\int_C f\,ds.
Vector line integral
An integral of a vector field along an oriented curve, usually written ∫CF⋅dr\int_C \mathbf{F}\cdot d\mathbf{r}, that measures work or circulation.
Orientation
The chosen direction of travel along a curve or the chosen normal direction on a surface.
Surface integral
An integral over a surface that uses an area element such as dSdS or an oriented vector area element.
Flux
The signed amount of a vector field passing through a surface, computed by ∬SF⋅n dS\iint_S \mathbf{F}\cdot\mathbf{n}\,dS.
Conservative field
A vector field F\mathbf{F} is conservative if F=∇f\mathbf{F}=\nabla f for some potential function ff, making line integrals path independent.

Common Mistakes to Avoid

  • Using dtdt instead of dsds in a scalar line integral is wrong because the arc length factor ∥r′(t)∥\|\mathbf{r}'(t)\| is required.
  • Forgetting orientation in ∫CF⋅dr\int_C \mathbf{F}\cdot d\mathbf{r} is wrong because reversing the curve changes the sign of the integral.
  • Using ∥ru×rv∥\|\mathbf{r}_u \times \mathbf{r}_v\| for flux is wrong when the surface is oriented because flux needs the signed vector area element ru×rv dA\mathbf{r}_u \times \mathbf{r}_v\,dA or its negative.
  • Applying Green’s theorem to a nonclosed curve is wrong because the theorem requires a simple closed boundary curve CC enclosing a region RR.
  • Using the divergence theorem on an open surface is wrong because the theorem applies to a closed surface bounding a solid region EE.

Practice Questions

  1. 1 Compute ∫C(x+y) ds\int_C (x+y)\,ds for the line segment parametrized by r(t)=⟨t,2t⟩\mathbf{r}(t)=\langle t,2t\rangle, 0≤t≤10\le t\le 1.
  2. 2 Evaluate ∫CF⋅dr\int_C \mathbf{F}\cdot d\mathbf{r} for F=⟨y,x⟩\mathbf{F}=\langle y,x\rangle and r(t)=⟨cos⁡t,sin⁡t⟩\mathbf{r}(t)=\langle \cos t,\sin t\rangle, 0≤t≤2π0\le t\le 2\pi.
  3. 3 Find the upward flux of F=⟨0,0,z⟩\mathbf{F}=\langle 0,0,z\rangle through the surface z=4−x2−y2z=4-x^2-y^2 over the disk x2+y2≤1x^2+y^2\le 1.
  4. 4 Explain when it is better to use Stokes’ theorem instead of directly computing ∮CF⋅dr\oint_C \mathbf{F}\cdot d\mathbf{r}, and state what orientation condition must match.

Understanding Line Integrals & Surface Integrals Reference

A useful first decision is to identify what is being accumulated. A scalar quantity has no direction. It might represent density on a wire, temperature along a path, or mass per unit area on a thin sheet.

The integral adds an amount based on the actual length or area of the object. A vector field has both size and direction. Its line integral measures how much the field acts along motion.

Its flux measures how much of the field passes through a surface. The dot product is important because it selects the part of the field that points along the path or through the surface.

A field perpendicular to motion does no work. A field tangent to a surface produces no flux through that surface.

Orientation carries physical information. For a path, choosing a travel direction determines whether a force helps or opposes the motion. Walking around the same loop in the other direction reverses the work or circulation result.

For a surface, a normal vector chooses one of two sides. For a closed surface, the standard choice is outward. This matches the idea of material leaving a container.

When a surface has a boundary, its orientation must agree with the direction chosen around that boundary. Use the right hand rule. Curl the fingers of your right hand in the boundary direction.

Your thumb points in the compatible normal direction. A sign error often comes from choosing a normal correctly at first, then using a boundary direction that does not match it.

The major theorems are not just shortcuts. They connect local behavior to a larger total effect. Green’s theorem relates circulation around a flat closed boundary to tiny rotational effects throughout its interior.

Stokes’ theorem gives the same idea for a curved surface in three dimensions. It says that the circulation around an edge depends on curl passing through any surface with that same edge. The divergence theorem relates outward flow through a closed boundary to sources and sinks inside the volume.

Positive divergence means that nearby field arrows spread outward overall. Negative divergence means they converge inward. These ideas appear in fluid flow, electric fields, heat transfer, and electromagnetic theory.

A sealed balloon with gas flowing outward across its surface has positive net flux. A fluid swirling around a drain shows circulation and curl.

Before applying a theorem, check its conditions carefully. The field must be defined and sufficiently smooth on the needed region. A hole, missing point, crack, or singularity can make a familiar shortcut invalid.

For example, a field that spins around the origin may have nonzero circulation around a loop even when its curl is zero away from the origin. The missing center matters. In calculations, sketch the curve or surface first.

Mark its orientation. State the parameter bounds. For surfaces, compute both tangent vectors before taking their cross product, then verify that the resulting normal points the intended way.

Test your answer with units. Work has units of force times distance, while flux has units of field strength times area. These simple checks catch many errors before the final integration.