why isn't my line integral equal from different paths?

Question

∫[x^2ydx + (x^2 - y^2)dy]

the first path is y=x^2 (0,0)(2,4).

so:
x = t
y = t^2
dx = dt
dy = 2tdt
0
∫(t^4 + 2t^3 - 2t^5)dt limits:{0,2}

the second path is y=2x. 0
so:
x=t
y=2t
dx = dt
dy = 2dt

so:
∫(2t^3 - 6t^2)dt limits:{0,2}

for the first one i get -104/15
and the second i get -8

they should be the same right?
did i mess up somewhere? or did you get the same answer?

Answer 1

Took me a while to trace through it, but I got the same answers as you did.

My calc book says that line integrals are independent of path "in a region D provided that given any two points A and B of D, the integral has the same value along every piecewise smooth curve or path in D from A to B".

When this happens, it's called a "conservative vector field".

I guess that is not a conservative vector field.

Answer 2

i did the calculations, u have no mistakes. U don t get the same result because u don t have a conservative field, as the other guy told u. To check it u look for a function f, for which gradf=.so, θf/θx=x^2 and θf/θy=x^2-y^2.if u integrate these two u ll see there is not such a function, which means the field is not a conservativ one and naturally u get different results.

Answer 3

No, they shouldn't be the same. You are integrating over 2 different paths, which gives you 2 different integrals.