Line Integrals Of Vector Fields

Question

How I can Evaluate (Line Integrals Of Vector Fields) ∫c (𝟑𝒙^𝟐−𝟑𝒚𝒛+𝟐𝒙𝒛) 𝒅𝒙 + (𝟑𝒚^𝟐−𝟑𝒙𝒛+𝒛^𝟐) 𝒅𝒚 + (𝟑𝒛^𝟐−𝟑𝒙𝒚+𝒙^𝟐+𝟐𝒚𝒛) 𝒅𝒛 along the line segment from (-1,2,3) to (3,2,-1)?

Answer 1

Consider f(x, y, z) = x^3 - 3xyz + x^2z + y^3 + yz^2 + z^3 and that we are integrating ∇f = F

Thus, the integral is

F(3, 2, 1) - F(-1, 2, 3) = 3^3 - 3*3*2*1 + 3^2*1 + 2^3 + 2*1^2 + 1^3 - (-1^3-3*-(-1)*2*3+(-1)^2*3+2^3+2*(3)^2+3^3) =

29 - 73 =

-44

Answer 2

Find the parametric equations of the line passing through the given points.

You should be getting

x = −1 + t

y = 2

z = 3 − t.

Obviously the interval of integration is 0 ≤ t ≤ 4.

then dx=dt , dy=0 and dz=−dt

Then substitute for x , y and z the corresponding values from the parametric equations

dx=dt , dy=0 and dz=−dt in the original integral then you will be getting an integral

in terms of t , with limits of integration from t = 0 to t= 4.