# 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)