Calculus 3 : Line Integrals of Vector Fields

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #1 : Line Integrals Of Vector Fields

Evaluate , where , and  is the curve given by .

 

Possible Answers:


Correct answer:

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 .

Possible Answers:

Correct answer:

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 #3 : Line Integrals Of Vector Fields

Evaluate  on the curve , where 

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

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

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