Line Integrals

The Essentials

Regular integrals are the sum of small bits of \( x \) multiplied by the value of the function at that small bit. Line integrals are integrated along small bits of the parameterization of a curve instead of the \( x \) axis. A line integral can be taken of a scalar function or a vector function. The first step to find a scalar or vector line integral is to find the parameterization vector of the line. The formula for scalar line integrals:

\[ \int_C fds=\int_a^b f(\vec{r}(t))||\vec{r'}(t)||dt \]

The formula for vector line integrals:

\[ \int_C \vec{F}\cdot \vec{T}ds=\int_C \vec{F}\cdot d\vec{r}=\int_a^b \vec{F}(\vec{r}(t))\cdot\vec{r'}(t)dt \]

Another way to denote vector line integrals is:

\[ \int_C \vec{F}\cdot d\vec{r}=\int_C Pdx+Qdy+Rdz \]

Where \( \vec{F}=\langle P,Q,R\rangle \) and \( d\vec{r}=\langle dx,dy,dz\rangle \).

A line integral where \( f \) is 1 will give the arc length of the line. Likewise when \( f \) is a density function, the line integral of \( f \) will give you the density of a wire. Work is a vector line integral.

Flux is fluid flow against a curve. It is the same as a vector line integral except that the unit tangent vector (\( \vec{T} \)) is replaced with the unit normal vector (\( \vec{N} \)). The normal vector is \( \vec{n}=\langle -y'(t),x'(t)\rangle \), so the unit normal vector is the normal vector divided by its magnitude:

\[ \vec{N}=\frac{\vec{n}}{||\vec{n}||} \]

Therefore the equation for flux is:

\[ \int_C \vec{F}\cdot\vec{N}ds=\int_C \vec{F}(\vec{r}(t))\cdot\vec{n}(t)dt \]

Circulation is the tendency of the curve to move in the same direction as the function. It is the vector line integral over a closed curve:

\[ \oint_C=\vec{F}\cdot\vec{T}ds \]

Example

Calculate the work done by the field \( \vec{F}(x,y)=\langle -y,x\rangle \) over the unit circle moving counterclockwise.

Work is a vector line integral. We can represent the parameterization as:

\[ \vec{r}(t)=\langle \cos(t),\sin(t)\rangle \]
\[ \vec{r'}(t)=\langle -\sin(t),\cos(t)\rangle \]

To calculate the work:

\[ W=\int_C \vec{F}\cdot d\vec{r}=\int_0^{2\pi}\langle -\sin(t),\cos(t)\rangle\cdot\langle-\sin(t),\cos(t)\rangle dt \]
\[ W=\int_0^{2\pi}\sin^2(t)+\cos^2(t) dt=2\pi \]

Practice

1) Calculate the mass of the wire that has a density function of \( \rho(x,y,z)=x^2+y^2+z^2 \) and a shape parameterized by \( \vec{r}(x,y,z)=\langle 3\cos(t),3\sin(t),t\rangle \) from \( 0\leq t\leq 3 \).

2) Evaluate the integral from the point (0, 1, 4) to (2, 5, 6):

\[ \int yzdx+xzdy+xydz \]

3) Calculate the flux of \( \vec{F}=x\hat{i}+y^2\hat{j} \) across the curve \( y=x \) from \( 0\leq x\leq 4 \)

Solutions:

1) \( 36\sqrt{10} \)

2) 60

3) \( -\frac{40}{3} \)

More Resources

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