Work - Line Integral

1. Jun 21, 2012

portuguese

1. The problem statement, all variables and given/known data

I have exam tomorrow and there's a problem I don't know how to do.

Consider the curve C1 (x=-y^2+3y) and C2 (x=0), both defined for y$\in$[0,3].
Calculate the work done by F(x,y)=(x,y^2) along the curve C=C1UC2 (retrograde direction).

2. Relevant equations

3. The attempt at a solution

The solution given is 0. I really don't know how to get it. Thanks!

2. Jun 21, 2012

Staff: Mentor

When you traverse the curve do you wind up at your staring point? if so then the work is zero by definition, right?

3. Jun 21, 2012

algebrat

This is not true in general. (If you have a conservative force then it is true; this force happens to be conservative, but how would you show that?)

You could parametrize the two separate trajectories as some r(t)=(x(t),y(t))

Then use ∫f(r(t))*r'(t)dt on each curve.

4. Jun 21, 2012

portuguese

How do I do it for C2? I think this is the biggest problem for me.

5. Jun 21, 2012

vela

Staff Emeritus
You need to show your work before you can receive help here.

6. Jun 21, 2012

algebrat

Hint: Organize your work, write C2:, and then write notes for that region, like r(t)=(0,y(t)), F(x,y)=F(0,y)=... You'll have to decide on the rest of the parametrization for r(t) is, that is, what do you think y(t) should be. Decide on a and b in t=a...b. Et cetera; good luck!

7. Jun 22, 2012

HallsofIvy

Staff Emeritus
As algebrat suggests, following on what jedishrfu said (hopefully jedishrfu had already noted, but forgot to say, that the force is conservative) the simplest way to do that is show that the force is conservative. portugese, do you know how to do that?