Calculate the line integral

where;

F(x, y, z) = (yz, xz, xy)
Υ(t)= (2 cos t, t^2 sin t, cos^2 t) tε [0; 3]
HINT: it is a conservative function

Question

Calculate the line integral

where;

 F(x, y, z) = (yz, xz, xy)     
Υ(t)= (2 cos t,  t^2 sin t, cos^2 t)     tε [0; 3]
HINT: it is a conservative function

anonymous · Answer

I believe they want you to solve some exact equations for this one, if they give you the hint that it's conservative, that means that the line integral will only depend on it's initial and end point.

So that will means that:
$$\int\limits_C yzdx+xzdy+xydz=\int\limits_a^b \nabla fdr$$

anonymous · Answer

Do you know how to setup your equations from here?

anonymous · Answer

I got f (x,y,z) = xyz , not sure about doing the integration....

anonymous · Answer

So you have found your potential function correctly, now you need to evaluate that function from the beginning point to the end point. You can also rewrite your potential function by substituting your given vector function.

anonymous · Answer

$$F(x,y,z)=xyz=2\cos (t) t^2 \sin(t) \cos^2t =2t^2 \cos^3(t)\sin(t)$$

anonymous · Answer

from t=0 to t=3

anonymous · Answer

I was told when the function is conservative it is a path independant.......but looks like for the final answer we still should Integrate the path by pluging the xyz value and using the point. right??

anonymous · Answer

No if you found your potential function then you just need to evalaute that function from the end and the start point. So if you have found your potential function, then there's is nothing much more to do than that.

anonymous · Answer

in this case, please let me know what you got for the final answer?

anonymous · Answer

18cos^3(3)sin(3)