INSTRUCTOR: Evaluate the line integral along the curve C of the function x squared plus yz, where C is the line with parameterization r of t equal to angle bracket 2t comma 5t comma negative t close angle bracket with 0 less than or equal to t less than or equal to 10. First, note that the parameterization gives us an x, a y, and a z value that we'll substitute into our integral. And because we have boundaries on t, you can set up an integral, as long as we note that the magnitude of r prime of t. We're going to have to multiply by a factor of that.

So let's go ahead and find the magnitude of r prime of t. Well, first [INAUDIBLE] the magnitude of the angle of the vector angle bracket 2 comma 5 comma negative 1. And that becomes the square root of 30. So we'll have to include a factor of that in our integral.

So for the first parameterization, we have f of t. This is going to be the integral from 0 to 10 of open bracket open parentheses 2t close parentheses squared plus open parentheses 5t closed parentheses open parentheses negative t close parentheses close bracket square root of 30 dt. Now, this isn't too complicated.

This becomes the integral from 0 to t-- or 0 to 10 with respect to t of the square root of 30-- sorry, of negative square root of 30 t squared, which will be equal to negative 1,000 times the square root of 30 all divided by 3. Now, reparameterize C with parameterization s of t equal to angle bracket 4t comma 10t comma negative 2t close angle bracket with 0 less than or equal to t less than or equal to 5 and recalculate the integral, line integral, along the curve C of the function x squared plus yz. And notice the change of parameterization had no effect on the value of the integral.

So let's do a similar process. We'll find the magnitude of s prime of t, which is the magnitude of the vector 4 comma 10 comma negative 2, which is equal to 2 square roots of 30. And now this line integral will be the integral from 0 to 5 with respect to t of the function open bracket open parentheses 4t close parentheses squared plus open parentheses 10t close parentheses open parentheses negative 2t close parentheses close bracket. And then we need to multiply by a factor of 2 square roots of 30.

Now, this becomes the integral from 0 to 5 with respect to t of the function-- let's see. That would be negative 8 square root of 30 multiplied by t squared. And that will evaluate to be negative 1,000 square roots of 30 all divided by 3.

Notice that the parameterization of r of t and s of t have the same beginning and ending point. The difference is s of t travels twice as fast, which makes sense that the line integral would in fact be the same. Reparameterization had no effect on the value of the integral.