Proving Path Independence: Evaluating the Integral on a Given Curve"

[joemama69]

Homework Statement

show that F is path independent. Then evaluate the integral F dot dr on c, where c = r(t) = (t+sin(pi)t) i + (2t + cos(pi)t) j, 0<=t<=1

Homework Equations

The Attempt at a Solution

F = 4x^3y^2 + 2xy^3 i + 2x^4y - 3x^2y^2 + 4y^3 j

grad f = 12x^2y^2 + 2y^3 i + 2x^4 - 6x^2y + 12y^2 j not sure i need this

my instructor talked about numerouse way to determine path independace. which is the easiest

 

[dx]

You mean the line integral of F is path independent? All you have to do is show that the curl of F is zero. Then the result follows from Stoke's theorem.

 

[joemama69]

ok so i found the curl of F
curl F = (8x^3y - 6xy^2 - 8x^3y + 6xy^2) = 0

but then the problem says to eval the integral F dot dr over the region c

when i dot them i got a extremely long expression. is this problem just a pain in the butt or did i make a boo boo

 

[dx]

You've shown that the line integral is path independent, so you can choose a more convenient path to do the integration. What does the curve C look like? What are its endpoints?