\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 227 Standard C11


Line Integrals

At the end of the course, each student should be able to…

  • C11: LineInt. Compute and apply line integrals.

C11. Line Integrals

  • The net area of the ribbon traveling along the curve \(C\) with height given by \(f(x,y,z)\) at each point is given by the line integral \(\int_C f\,ds\).
    • By parametrizing \(C\) with \(\vect r(t),a\leq t\leq b\) and computing \(\frac{ds}{dt}\), this integral may be evaluated by \(\int_a^b f(\vect r(t))\frac{ds}{dt}\,dt\).
  • The work done by the force vector field \(\vect{F}\) along the curve \(C\) is given by the line integral \(\int_C\vect F \cdot\vect {T}\,ds\) where \(\vect T\) is the vector field of unit tangent vectors at each point of \(C\).
    • This integral is often written as \(\int_C\vect F\cdot d\vect{r}\).
    • By parametrizing \(C\) with \(\vect r(t),a\leq t\leq b\) and computing \(\frac{d\vect{r}}{dt}\), this integral may be evaluated by \(\int_a^b \vect F(\vect r(t))\cdot\frac{d\vect{r}}{dt}\,dt\).
    • This integral is often expanded as \(\int_C\<M,N,P\>\cdot\<dx,dy,dz\>= \int_C M\,dx + \int_C N\,dy + \int_C P\,dz\) \(= \int_a^b M\frac{dx}{dt}\,dt + \int_a^b N\frac{dy}{dt}\,dt + \int_a^b P\frac{dz}{dt}\,dt\)
    • The flow of a vector field through a curve is also given by \(\int_C\vect F\cdot\vect T\,ds\).
  • The flux of the planar vector field \(\vect{F}\) moving across the closed planar curve \(C\) is given by the line integral \(\int_C\vect F \cdot\vect {n}\,ds\) where \(\vect{n}\) is the vector field of outward normal unit vectors at each point of \(C\).
    • This integral may be evaluated as \(\int_C M\,dy-\int_C N\,dx\), assuming counter-clockwise motion on \(C\).
  • The usual rules of additivity, constant multiples, and so on apply.

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 15.1 (exercises 9-32)
    • 15.2 (exercises 7-34)