Calculus 3 : Line Integrals of Vector Fields

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 .

The formula for work is given by

Writing our path in parametric equation form, we have

Plugging this into our work equation, we get

Example Question #3 : Line Integrals Of Vector Fields

Evaluate on the curve , , where .

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

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

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

University of Science and Arts of Oklahoma, Bachelor of Science, Mathematics. Florida Institute of Technology, Master of Scie.

Pennsylvania State University-Main Campus, Bachelor of Science, Chemical Engineering. Carnegie Mellon University, Doctor of P.

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