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)