Line integral over a given curve C

[nmelott]

Homework Statement

Evaluate the line integral over the curve http://webwork.math.ttu.edu/webwork2_files/tmp/equations/33/92ca0e3b907f876e5a974ad1457d1f1.png from

to http://webwork.math.ttu.edu/webwork2_files/tmp/equations/6f/bccf9dd59b9c22450c590a042bb77d1.png .

∫-ydx+3xdy (over the curve C)

Homework Equations

The Attempt at a Solution

I'm really stuck on this problem not doing very well with line integrals.
I started by changing y^2=x to parametric --> x=t^4 y=t^2
Then I took the derivate of each one --> dx=4t^3dt dy=2tdt
I then plugged in each term into the given integral --> ∫-ydx+3xdy (over the curve C) = ∫-(t^2)(4t^3)+3(t^4)(2t)dt

Would I use the points given to get my limits of integration or am I way off?

 

[Mark44]

Homework Statement

Evaluate the line integral over the curve http://webwork.math.ttu.edu/webwork2_files/tmp/equations/33/92ca0e3b907f876e5a974ad1457d1f1.png from

to http://webwork.math.ttu.edu/webwork2_files/tmp/equations/6f/bccf9dd59b9c22450c590a042bb77d1.png .

∫-ydx+3xdy (over the curve C)

Homework Equations

The Attempt at a Solution

I'm really stuck on this problem not doing very well with line integrals.
I started by changing y^2=x to parametric --> x=t^4 y=t^2

A simpler set would be x = t², y = t. You can use the given points on C to figure out the interval for t values.

Then I took the derivate of each one --> dx=4t^3dt dy=2tdt
I then plugged in each term into the given integral --> ∫-ydx+3xdy (over the curve C) = ∫-(t^2)(4t^3)+3(t^4)(2t)dt

Would I use the points given to get my limits of integration or am I way off?

 

[nmelott]

Got it,
Thank you!

 

[STEMucator]

You could have just chosen $x = y^2$ and $y = y$ (where $y$ is the parameter). Then you can see that $1 \leq y \leq 3$.

Then computing $\frac{dx}{dy} = 2y$ will give you $dx = 2y dy$.

Subbing everything in you should find the same answer.