Section 5.2 Line Integral

5.2 Line Integral

A line integral gives us the ability to integrate multivariable functions and vector fields over arbitrary curves in a plane or in space. There are two types of line integrals: scalar line integrals and vector line integrals. Scalar line integrals are integrals of a scalar function over a curve in a plane or in space. Vector line integrals are integrals of a vector field over a curve in a plane or in space. Let’s look at scalar line integrals first. 

Definition 

Let $f$ be a function with a domain that includes the smooth curve $C$ that is parameterized by $\overrightarrow{r(t)}=<x(t),y(t),z(t)>$, $a\le t\le b$. The scalar line integral of $f$ along $C$ is 

$${\int }_{C}f(x,y,z)ds=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}f(P_i^{\ast })\mathrm{△}{s}_i$$

if this limit exists. 

If $C$ is a planar curve, then $C$ can be represented by the parametric equations $x=x(t),y=y(t)$, and $a\le t\le b$. If $C$ is smooth and $f(x,y)$ is a function of two variables, then the scalar line integral of $f$ along $C$ is defined similarly.

Theorem: Evaluating a Scalar Line Integral 

Let $f$ be a continuous function with a domain that includes the smooth curve $C$ with parameterization $\overrightarrow{r(t)}$, $a\le t\le b.$ Then

$${\int }_{C}f(x,y,z)ds={\int }_a^{b}f(\overrightarrow{r(t)})||\overrightarrow{r^{\prime }(t)}||dt.$$

Theorem: Scalar Line Integral Calculation 

Let $f$ be a continuous function with a domain that includes the smooth curve $C$ with parameterization $\overrightarrow{r(t)}=<x(t),y(t),z(t)>$, $a\le t\le b$. Then

$${\int }_{C}f(x,y,z)ds={\int }_a^{b}f(\overrightarrow{r(t)})\sqrt{(x^{\prime }(t){)}^2+(y^{\prime }(t){)}^2+(z^{\prime }(t){)}^2}dt.$$

Similarly, 

$${\int }_{C}f(x,y)ds={\int }_a^{b}f(\overrightarrow{r(t)})\sqrt{(x^{\prime }(t){)}^2+(y^{\prime }(t){)}^2}dt$$

if $C$ is a planar curve and $f$ is a function of two variables. 

Example 1: Evaluate ${\int }_{C}yds$ where $C$ is the curve $\overrightarrow{r(t)}=<t^2,2t>$ with $0\le t\le 2$.

Exercise 1: Evaluate ${\int }_{C}xds$ where $C$ is the curve $\overrightarrow{r(t)}=<3t,t^2>$ with $0\le t\le 1$. 

Example 2: Evaluate ${\int }_C{e}^{y}ds$ where $C$ is the line segment from $(1,2)$ to $(3,4)$.

Exercise 2: Evaluate ${\int }_C{e}^{x}ds$ where $C$ is the line segment from $(0,1)$ to $(-1,2)$. 

Example 3: Evaluate ${\int }_C{x}^2+y^2-zds$ where $C$ is the part of the helix $\overrightarrow{r(t)}=<\text{cos}(3t),\text{sin}(3t),t>$, $0\le t\le \pi$.

Exercise 3: Evaluate ${\int }_C{x}^2+z^2+yds$ where $C$ is the part of the helix $\overrightarrow{r(t)}=<\text{cos}(2t),t,\text{sin}(2t)>$, $0\le t\le \frac{\pi }{2}$.

Example 4: A wire has a shape that can be modeled with the parameterization $\overrightarrow{r(t)}=<\text{cos}(t),\text{sin}(t),t>$, $0\le t\le 4\pi$. Find the length of the wire.

Exercise 4: A wire has a shape that can be modeled with the parameterization $\overrightarrow{r(t)}=<\text{sin}(t),2t,\text{cos}(t)>$, $0\le t\le \pi$. Find the length of the wire.

Definition: line integral with respect to arc length

Let $C$ be the curve $\overrightarrow{r(t)}=<x(t),y(t),z(t)>$ then we can define the line integral with resopect to each component of $\overrightarrow{r(t)}$. 

$${\int }_{C}f(x,y,z)dx={\int }_a^{b}f(\overrightarrow{r(t)})x^{\prime }(t)dt$$

$${\int }_{C}f(x,y,z)dy={\int }_a^{b}f(\overrightarrow{r(t)})y^{\prime }(t)dt$$

$${\int }_{C}f(x,y,z)dz={\int }_a^{b}f(\overrightarrow{r(t)})z^{\prime }(t)dt.$$

Example 5: Evaluate ${\int }_{C}xy+\text{cos}(x)dx$ where $C$ is the arc of $x=y^2$ from $(0,0)$ to $({\pi }^2,\pi )$. 

Exercise 5: Evaluate ${\int }_C{x}^{2}y+\text{sin}(x)dy$ where $C$ is the arc of $y=x^2$ from $(0,0)$ to $(\pi ,{\pi }^2)$. 

Definition: We define the integral of $F$ along $C$ as the sum of the integrals of $f$ along each of the smooth pieces of $C$:

$${\int }_{C}f(x,y,z)ds={\int }_{C_1}f(x,y,z)ds+{\int }_{C_2}f(x,y,z)ds+\cdots +{\int }_{C_n}f(x,y,z)ds.$$

Example 6: Evaluate ${\int }_C{x}^{2}dx+y^{2}dy$ where $C$ is the arc of circle $x^2+y^2=9$ from $(0,3)$ to $(-3,0)$ followed by the line segment from $(-3,0)$ to $(-4,-5)$. 

Exercise 6: Evaluate ${\int }_{C}ydx+xdy$ where $C$ is the arc of circle $x^2+y^2=1$ from $(1,0)$ to $(0,1)$ followed by the line segment from $(0,1)$ to $(-1,2)$. 

Example 7: Evaluate ${\int }_C{z}^{2}dx+x^{2}dy+y^{2}dz$ where $C$ is the line segments from $(1,0,0)$ to $(1,0,2)$ and from $(1,0,2)$ to $(2,3,4)$.

Group work:

1. Evaluate ${\int }_C{x}^2+yzds$ where $C$ is the curve $\overrightarrow{r(t)}=<2t,5t,-t>$, $0\le t\le 4$.

2. Evaluate ${\int }_{C}xyds$ where $C$ is the first half of the circle $x^2+y^2=9$ and is traverse in the clockwise direction.

3. Evaluate ${\int }_C{y}^{2}dx+xdy$ where $C$ is the arc of $x=4-y^2$ from $(4,0)$ to $(0,2)$.