Calculus 3 Line integral

Question

Consider the vector field  Evaluate  along the curve : 

Answer 1

APPROACH ONE USING THE DEFINITION

The ∫_CF • d r = ∫^t=2_t=0 < 3 t² + 4 t³ , 4 t² + 6 t³ > • < 2 t, 3 t² > d t =

∫^t=2_t=0 [ 20 t⁴ + 6 t³ + 18t⁵] d t = 344.

APPROACH TWO USING THE FUNDAMENTAL THEOREM FOR LINE INTEGRALS

Let us find the potential function of the given vector field.

The given vector field is a conservative one

∂ P / ∂ y = ∂ / ∂y [ 3 x +4 y ] = 4 and ∂ Q / ∂ x = ∂ / ∂ x [4 x +6 y ]= 4.

That i s ƒ[x, y] = ∫ [ 3x + 4y] d x = 3x²/2 +4xy +h[y]

∂ ƒ/ ∂ y = 4x + h' [ y ] = 4x + 6y ⇒ h' [ y ] = + 6y ⇒ h[y] = 3y²

ƒ[x, y] = = 3x²/2 +4xy +3y² . ∇ ƒ = <3x+4y,4x+6y>

Then ∫_CF • d r = ∫_C ( ∇ ƒ ) • d r = ∫₀² d/ dt{ ƒ[r(t)]} d t=

ƒ[r(2)] - ƒ [r(0)] =3/2 4² + (4 ) ( 4) ( 8) + 3( 64) = 344