# Calculus 3 : Line Integrals

## Example Questions

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### Example Question #1 : Line Integrals

Write the parametric equations of the line that passes through the points  and .

Explanation:

First, you must find the vector that is parallel to the line.

This vector is

.

From the points we were given, this becomes

.

To form the parametric equations, we need to pick a point that lies on the line we want.

The point  is used.

The vector form of the line is from the following equation

.

We then rewrite each expression in terms of the variables x, y, and z.

### Example Question #2 : Line Integrals

Evaluate the line integral of the function

over the line segment  from  to

Explanation:

Evaluate the line integral using the function

over the line segment  from  to

Define the Parametric Equations to Represent

The points given lie on the line . Define the parameter , then  can be written . Therefore, the parametric equations for  are:

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The line integral of a function  along the curve  with the parametric equation  and  with  is defined by:

(1)

Where  is the vector derivative of the vector , therefore  is simply the magnitude of the vector derivative.

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Write the vector :

Differentiate,

The absolute value (magnitude) of this vector is:

Write the function  in terms of the parameter

Insert everything into Equation (1) noting that the limits of integration will be  due to the fact that the parameter  varies from  to  over the line segment we are integrating over.

### Example Question #3 : Line Integrals

Evaluate , where , and  is any path that starts at , and ends at .

Explanation:

Since there isn't a specific path we need to take, we just evaluate  at the end points.

### Example Question #1 : Green's Theorem

Use Green's Theorem to evaluate , where  is a triangle with vertices  with positive orientation.

Explanation:

First we need to make sure that the conditions for Green's Theorem are met.

The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).

In this particular case , and , where , and  refer to .

We know from Green's Theorem that

So lets find the partial derivatives.

### Example Question #2 : Green's Theorem

Use Green's Theorem to evaluate the line integral

over the region R, described by connecting the points , orientated clockwise.

Explanation:

Using Green's theorem

since the region is oriented clockwise, we would have

which gives us

### Example Question #3 : Green's Theorem

Use Greens Theorem to evaluate the line integral

over the region connecting the points  oriented clockwise

Explanation:

Using Green's theorem

Since the region is oriented clockwise

### Example Question #1 : Line Integrals

Compute  for

Explanation:

In order to find the divergence, we need to remember the formula.

Divergence Formula:

, where , and  correspond to the components of a given vector field .

Now lets apply this to our situation.

### Example Question #5 : Line Integrals

Compute  for

Explanation:

In order to find the divergence, we need to remember the formula.

Divergence Formula:

, where , and  correspond to the components of a given vector field .

Now lets apply this to our situation.

### Example Question #2 : Line Integrals

Compute  for

Explanation:

In order to find the divergence, we need to remember the formula.

Divergence Formula:

, where , and  correspond to the components of a given vector field .

Now lets apply this to our situation.

### Example Question #7 : Line Integrals

Find , where

Explanation:

In order to find the divergence, we need to remember the formula.

Divergence Formula:

, where , and  correspond to the components of a given vector field .

Now lets apply this to our situation.

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