All Calculus 3 Resources
Example Question #1 : Line Integrals Of Vector Fields
Evaluate , where , and is the curve given by , .
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