Line Integral and Vector Field Problem

[zm500]

Homework Statement

Find the work done by the force field F(x,y) = x sin(y)i + yj on a particle that moves along on the parabola y = x^2 from (-1,1) to (2,4).

Homework Equations

Work = line integral of the dot product of Field vector and change in the path
The path is parabola equation.

The Attempt at a Solution

I tried to integrate with respect to x and y instead of t, because I don't know how to find the path vector r(t).
So, i got two integrals for X-direction and y - direction.
Then, I can just use the Pythagorean theorem to find the total work.

Srry, I didn't write any mathematical equations.

 

[HallsofIvy]

The "path vector" is always a position vector:
$$\vec{r}(x)= x\vec{i}+ y\vec{j}= x\vec{i}+ x^2\vec{j}$$
so that
$$d\vec{r}= (\vec{i}+ 2x\vec{j})dx$$

If the force field is not conservative, the work done to move along two legs of a right triangle might have nothing to do with the work required to move along the hypotenuse. I don't believe that using the "Pythagorean Theorem" will work here.

 

[zm500]

The "path vector" is always a position vector:
$$\vec{r}(x)= x\vec{i}+ y\vec{j}= x\vec{i}+ x^2\vec{j}$$
so that
$$d\vec{r}= (\vec{i}+ 2x\vec{j})dx$$

If the force field is not conservative, the work done to move along two legs of a right triangle might have nothing to do with the work required to move along the hypotenuse. I don't believe that using the "Pythagorean Theorem" will work here.

Thanks!