Calculus 2 Line integrals

Question

Consider the vector field 

Use your answer to evaluate 

 along the curve C: 

Answer 1

Consider the vector field F=<x^3*y^4,x^4*y^3> = < P [x ,y] ,Q [x , y ] >

Use your answer to evaluate 

∫CF⋅dr

 along the curve C: 

r(t)=6cos(t)i+6sin(t)j, 0≤t≤π4

∂ P / ∂ y = ∂ Q / ∂ x = 4 x³ y³ hence the vector field is a conservative one.

Therefore ∃ a function ƒ(x , y ) such that ∇ƒ = F ( x, y )

ƒ(x, y ) = ∫ x³ y⁴ d x = (1/4) x⁴ y⁴ + h ( y )

∂ ƒ / ∂ y = x⁴ y³ + h' ( y ) =x⁴ y³ ⇒ h' ( y ) = 0 ⇒ h ( y ) = ξ , where ξ is a constant.

Then ∫_CF• d r = ∫₀^π/4 ∇ƒ( r(t)) • r'(t) d t = ƒ( r(π/4)) - ƒ( r(0)) = 26244

Answer 2

It is not too hard to guess the form of a potential field V = (1/4) x⁴ y⁴

If the upper limit on t is pi/4 (not pi 4) ,

the starting point is (6 ,0) and the ending point is ( 6 sqrt(2) /2 , 6 sqrt(2)/2 )

The value of the contour integral is thus 6⁸ / 64