# Calculus 3 : Line Integrals of Vector Fields

## Example Questions

### Example Question #1 : Line Integrals Of Vector Fields

Evaluate , where , and  is the curve given by .

Explanation:

First we need to evaluate the vector field evaluated along the curve.

Now we need to find the derivative of

Now we can do the product of  and .

Now we can put this into the integral and evaluate it.

### Example Question #2 : Line Integrals Of Vector Fields

Find the work done by a particle moving in a force field , moving from  to  on the path given by .

Explanation:

The formula for work is given by

.

Writing our path in parametric equation form, we have

.

Hence

Plugging this into our work equation, we get

.

### Example Question #2 : Line Integrals Of Vector Fields

Evaluate  on the curve , where

Explanation:

The line integral of a vector field is given by

So, we must evaluate the vector field on the curve:

Then, we take the derivative of the curve with respect to t:

Taking the dot product of these two vectors, we get

This is the integrand of our integral. Integrating, we get

### Example Question #4 : Line Integrals Of Vector Fields

Evaluate the integral  on the curve , where , on the interval

Explanation:

The line integral of the vector field is equal to

The parameterization (using the corresponding elements of the curve) of the vector field is

The derivative of the parametric curve is

Taking the dot product of the two vectors, we get

Integrating this with respect to t on the given interval, we get

### Example Question #5 : Line Integrals Of Vector Fields

Calculate  on the interval , where  and

Explanation:

To calculate the line integral of the vector field, we must evaluate the vector field on the curve, take the derivative of the curve, and integrate the dot product on the given interval.

The vector field evaluated on the given curve is

The derivative of the curve is given by

The dot product of these is

Integrating this over our given t interval, we get