the fundamentam theorem of line integrals

[jakeisthesnake]

Hello,

So I am having a bit of difficulty solving the following problem:

Evaluate the following integral using the fundamental theorem of line integrals: ?(0,0,0)to(?/2,3,4) -sinx dx +z dy +y dz

What I have so far:

I define the verctor F = (-sinx)i + (z)j + (y)k which has the form F= Ni + Mj+ Pk
It is conservative because dN/dy=dM/dx, dN/dz=dP/dx, and dM/dz=dP/dy.

So I want U(x,y,z) to be the function such that ?a to b F dR= U(b)-U(a).

The formula is dU/dx = N, dU/dy=M and dU/dz = P

so I start with dU=N dx
U = ?-sin x dx

U = cos x + h(y) + g(z)
then is partially differentiate with respect to y to get
dU/dy = h'(y)
but since we know that dU/dy = M
we have
h'(y) = z
therefore h(y) = zy

substituing we have
U = cos x + zy + g(z)
a by a similar process we get
dU/dz = y + g'(z)= y = P
thus g'(z)= 0
and g(z) = C

leaving U = cos x + zy + C
pluging in our points (0,0,0) and (?/2,3,4) we get 13 which is unfortunately incorrect.

I believe that my error lies somewhere in the fact the U should have been
U= cos x + h(y,z)
or something like that in which case I am not sure how to continue?

would dU/dy = dh/dy qnd from there how would i use that info?

Thanks!

 

[jakeisthesnake]

oops so I just noticed that substituting the pointing into U is
U(pi/2,3,4)-U(0,0,0)
=(cos pi/2 + 12)- (cos 0 + 0)
= 0 +12 -1 -0
= 11
which is the answer
but I would still be grateful in anyone can confirm that I correctly took the integral.

Thanks