Integrating along paths

1. Feb 5, 2012

Sekonda

Hey guys,

I have :

df=c(x^2)(y^2)dx + (x^3)(y)dy

along paths (0,0) to (1,1); and also paths (0,0) to (0,1) to (1,1) (where (x,y))

where c is some constant.

I am having difficulty doing this particular integral, what type of integral is it and how do I go about solving it?

Thanks!

2. Feb 5, 2012

Clever-Name

It's just a line integral. It's just asking you to do the same integral in two different ways (along two different paths). For the first case, can you think of a way to parametrize the path into a single variable? For the second case, think about what x, y, dx, and dy are between each point.

3. Feb 5, 2012

Sekonda

So would I use parameters of x=sint, y=-cost for 0<t<pi for path (0,0) to (1,1)

and for the path (0,0) to (0,1) to (1,1) the fact that dx is zero for the first path and dy is zero for the second path?

4. Feb 5, 2012

Sekonda

I managed to attain (2^0.5)(c+1) and c/3 as my answers however I am not convinced that these are correct. For the path (0,0) to (1,1) I used parameterization : x=sin(t) y=sin(t) for 0<t<pi/2

Is this correct?

5. Feb 5, 2012

Sekonda

I just used x=y for the first path to attain a seemingly more likely answer of (c+1)/5, now I'm just stuck on the second path!

6. Feb 5, 2012

Sekonda

For the second path (0,0) to (0,1) to (1,1) I attained 2(c+1)/5, for the first path (0,0) to (1,1) I attained (c+1)/5.

Is this right?

7. Feb 5, 2012

vela

Staff Emeritus
This parameterization won't work because (x(t), y(t)) doesn't pass through (1,1).

Yes.

That parameterization will work, but you didn't get the right result. For the second path, the answer is indeed c/3.

That's right.