Math 252- 15.2: Line Integrals

Front 1

What is a line integral?

Back 1

is an integral where you evaluate over a curve.

Front 2

How to parameterize the line segment:

Back 2

1. figure out what variable to use as the parameter "t" as well as start & end values for the parameter
- start & end points are arbitrary

2. have to come up w/ an equation describing the relationship between x w/ respect to t.
- start by drawing up a chart & then plot these pts. from the chart onto an xt-plane.

3. find the equation of the line represented in the diagram
- will give us the relationship between t & x. Want eq. in x=mt+b

4. Look at the x & y coord. only to parameterize y.
- sketch points on xy-plane and fine y=mx+b sub x eq. relating to t.

5. to parameterize z, look at only yz-plane find slope and plug in to get eq. z-z1=m(y-y1)
- sub parameterization for y inot eq. to get a z t relation.

6. final answer written as a vector-value funciton r(t)=

Front 3

Constructing a piecewise smooth parameterization:

Back 3

If C consits of line segments such as C1,C2, C3, you can construct a smooth parameterization for each segment and piece them together by making the last t-value in Ci correspond to the first t-value in Ci+1

Front 4

Evaluation of a line integral as a definite integral:

Back 4

Front 5

Procedure to evaluate type #1 line integrals which are used to represent the mass of a wire where f(x,y,z) is the density funciton

Back 5

1. parameterize the curve r(t)

2. evaluate expression "ds"= ||r'(t)||

3. subsitute components of r(t) into the integrand & solve.

Front 6

How would you figure out the length of curve C?

Back 6

Front 7

Definition of the line integral of a vector field:

Back 7

Front 8

Procedure to evaluate type #2 line integrals which are used to pepresent either the work done by an object as it moves along the curve subject to the force give, or the flow of a fluid along a curve where F represents a velocity field instead of a force field

Back 8

1. Parameterize the curve if it is not already. Be sure to determine the range for the paramenter t. That is, determine a<t<b.

2. Evaluate F on the curve as a function of the parameter t.

3. find dr/dt

4. dot F with dr/dt

5. integrate from t=a to t=b

Front 9

For line integrals of vector functions, the orientation of the curve C is important. If the orientation of the curve is reversed, what occurs?

Back 9

The unit tangent vector T(t) is changed to -T(t), and you obtain the integral as followed:

Front 10

How do you find work when dealing with more than 1 curve?

Back 10

1. have to separately parameterize the curves r(t)
- the work done will be the sum of the separate line integrals

2. write the sum of the two integrals, F(x,y,z) needs to be separately evaluated for each curve.

3. find r'(t)

4. finally set up integraiton and solve.

5. sum up the answers for all the integrations to get final answer.

Front 11

Procedure to evaluate type #3 line integrals known as the "differential form" for a line integral. It is also used to represent the work done by a field or the low of a fluid along a curve.

Back 11

1. Parameterize the curve
- curve represented by the vector-valued funciton r(t).

2. find dx & dy with respect to T

3. Plug in to integral and solve.

Front 12

To go in the opposite direction, we could leave the parameterization the same except have the parameter t start at b and end at a.

Back 12