LINE INTEGRAL OF A CONSERVATIVE VECTOR FIELD.

YOU WILL LEARN ABOUT:

How to find the line integral of functions of two or three variables and of vector fields along a smooth curve.

If $C$ is a smooth curve divided into $n$ subarcs of length $∆ s$ , and $f$ is a function of $x$ and $y$ defined on $C$ , then the line integral of $f$ along $C$ is

∫_{C} f (x, y) d s = \lim_{n \to \infty} \sum_{i = 1}^{n} f (x_{i}^{*}, y_{i}^{*}) ∆ s_{i}

provided that the limit exists. If $C$ is given by the parametric equations

x = x (t), y = y (t), a \leq t \leq b

then

∫_{C} f (x, y) d s = ∫_{a}^{b} f (x (t), y (t)) \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t .

Line integrals can be similarly defined for functions of three variables.

If $C$ is a smooth curve divided into $n$ subarcs of length $∆ s$ , and $F$ is a continuous vector field defined on $C$ given by the vector function $r (t) = x (t) i + y (t) j + z (t) k$ on $a \leq t \leq b$ , then the line integral of $F$ along $C$ is

∫_{C} F \cdot d r = ∫_{C} F \cdot T d s = ∫_{a}^{b} F (r (t)) \cdot r' (t) d t,

where $T$ is the unit tangent vector at the point $(x, y, z)$ .

Line integrals can be used to determine many important physical quantities, such as the following.

Meteorologists can use line integrals to measure circulation along a closed curve, such as the circular flow of the wind path of a hurricane or tornado.

An electronics manufacturer can use line integrals to find the surface area of a curved TV's display, in order to determine the exact amount of glass needed for the screen. This information can be used to determine the exact cost of manufacturing the screen.

A surveyor can use line integrals to determine the average elevation of a winding downhill road.

Line Integral of a Conservative Vector Field

EQUATIONS

n \to \infty

Number of rectangles

4

Integral 1

Integral 2

Integral 3

\int_{c} f (x, y) d s = \int_{c} f (x (t), y (t)) \sqrt{({\frac{d x}{d t})}^{2} + ({\frac{d y}{d t})}^{2}} d t = 8.26895

Difference

= 0.32436

Σ_{i = 1}^{n} f (x_{i}^{*}, y_{i}^{*}) Δ s_{i} = 7.94459

Arc length

= \int_{c} \sqrt{({\frac{d x}{d t})}^{2} + ({\frac{d y}{d t})}^{2}} d t = 3.95278

\int_{c_{1}} f (x, y) d s = 1.532

\int_{c_{2}} f (x, y) d s = 3.167

\int_{c_{3}} f (x, y) d s = 2.583

\int_{c} f (x, y) d s = 7.282

\int_{c} f (x, y, z) d s = \int_{0}^{0} \sqrt{({\frac{d x}{d t})}^{2} + ({\frac{d y}{d t})}^{2} + ({\frac{d z}{d t})}^{2}} d t = 0.00

INSTRUCTIONS

In this example, $C = {(x, y) : x = t,$ $y = \sin (2 t),$ $0 \leq t \leq \frac{3 π}{4}}$ and
$f (x (t), y (t)) = 3 e^{- \frac{t}{3}}$ .

Use the slider to change the number of approximating rectangles to answer the following questions. You can rotate the graph to get a different view of the approximations.

PRACTICE QUESTIONS

INSTRUCTIONS

The curve $C$ to the left is the union of the following three smooth curves. $C_{1} = {(x, y) : x = t, y = t,$ $0 \leq t \leq 1}$ $C_{2} = {(x, y) : x = 2 t - 1,$ $y = 1,$ $1 \leq t \leq 2}$ $C_{3} = {(x, y) : x = 3,$
$y = t - 1,$ $2 \leq t \leq 3}$

PRACTICE QUESTIONS

INSTRUCTIONS

The curve to the left shows the path $C$ of a fly after it takes off from the top of a table. Consider two parameterizations of $C -$ one representing a "slow" traveling fly and one representing a fly traveling twice as fast as the "slow" fly.

YOU WILL LEARN ABOUT:

INSTRUCTIONS

In this example, C={(x,y):x=t, y=sin(2t), 0≤t≤3π4} and f(x(t),y(t))=3e−t3.

Use the slider to change the number of approximating rectangles to answer the following questions. You can rotate the graph to get a different view of the approximations.

PRACTICE QUESTIONS

INSTRUCTIONS

The curve C to the left is the union of the following three smooth curves. C1={(x,y):x=t,y=t, 0≤t≤1} C2={(x,y):x=2t−1, y=1, 1≤t≤2} C3={(x,y):x=3, y=t−1, 2≤t≤3}

PRACTICE QUESTIONS

INSTRUCTIONS

The curve to the left shows the path C of a fly after it takes off from the top of a table. Consider two parameterizations of C- one representing a "slow" traveling fly and one representing a fly traveling twice as fast as the "slow" fly.

PRACTICE QUESTIONS

In this example, $C = {(x, y) : x = t,$ $y = \sin (2 t),$ $0 \leq t \leq \frac{3 π}{4}}$ and
$f (x (t), y (t)) = 3 e^{- \frac{t}{3}}$ .

The curve $C$ to the left is the union of the following three smooth curves. $C_{1} = {(x, y) : x = t, y = t,$ $0 \leq t \leq 1}$ $C_{2} = {(x, y) : x = 2 t - 1,$ $y = 1,$ $1 \leq t \leq 2}$ $C_{3} = {(x, y) : x = 3,$
$y = t - 1,$ $2 \leq t \leq 3}$

The curve to the left shows the path $C$ of a fly after it takes off from the top of a table. Consider two parameterizations of $C -$ one representing a "slow" traveling fly and one representing a fly traveling twice as fast as the "slow" fly.