Parametric Equations for Line Integral: Finding the Correct Solution for C2

[Puchinita5]

Evaluate the line integral by two methods: A) directly and B) using Green's Theorem.

\oint xydx +x^2y^3dy
where C is the triangle with vertices (0,0) , (1,0), and (1,2).

I don't need the whole problem done, but I need someone to show me the work for finding the parametric equations for part A because I am not getting the same answer as in the book.

Basically, the part I'm getting wrong is the parametric equations for C2, or the vertical line on the right side of the triangle.

I put that r=(1-t)<1,0>+(t)<1,2> = <1-t, 0> + <t, 2t> = <1, 2t>

so x=1, y=2t...

but my solutions manual says y=t. And I looked this problem up on Cramster and it said the same thing.

Why does y=t and not 2t? Where am I messing up?

 

[ArcanaNoir]

Stewart Calculus? If so, what chapter/problem?

 

[Puchinita5]

Chapter 17.4 problem 3 in the 6th edition.

 

[stringy]

A line can be parameterized many different ways. What do they have t running over? From 0 to 2? Are you having t run from 0 to 1?

 

[ArcanaNoir]

Ah. They're running t between 0 and 2. It's the same as what you have (2t) between 0 and 1. Try doing it their way and see if you get the right answer. You should get the same answer your way if you make any necessary adjustments. Can't think of what adjustments they would be.. I think you just adjust the limits on your integral. I did this chapter ...five weeks ago now. My, how quickly I forget...

 

[Puchinita5]

Hmmm. Now that you mention that, they have t going from 0 to 2 and so I would get the same answer since I was using 0 to 1. So I guess it doesn't matter?

 

[stringy]

Yup. The interval over which the parameter ranges is just as important as the equation defining the parameterization!

And yes, if you do it with your parameterization you'll get the correct answer.

 

[ArcanaNoir]

What really matters is have you gotten the correct answer now?

 

[Puchinita5]

Thanks! I was worried that since this is the first time i saw paremtrization in a few weeks that I had forgotten how to do them already. I was so confused!

I feel better now :)