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Line Integrals & Surface Integrals Reference cheat sheet - grade college

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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
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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.