# Math Help - Line Integral

1. ## Line Integral

c(t) = (cos^3(t),sin^3(t)) t goes from 0 to 2pi
F(x,y) = xi + yj

I know the equations but something tells me if i do the gradient it equals 0. The problem I'm having is how to do and find the gradient.

2. you still need to say that you want integrated

maybe $\int_C F\cdot dC$

3. Why gradient? That is the opposite of the integral. I presume that you mean you are to integrate $\int_C F(x,y)\cdot d\vec{s}= \int_C xdx+ ydy$. If $C(t)= cos^3(t)\vec{i}+ sin^3(t)\vec{j}$ then $d\vec{s}= -3sin(t)cos^2(t)dt\vec{i}+ 3sin^2(t)cos(t)dt\vec{j}$ or $dx= -3sin(t)cos^2(t)dt$ and $dy= 3sin^2(t)cos(t)dt$

$F(x,y)= cos^3(t)\vec{i}+ sin^3(t)\vec{j}$ so you integral is
$\int_0^{2\pi} (-3sin(t)cos^5(t)+ 3sin^5(t)cos(t))dt$

Since that involves only odd powers of sine and cosine, it should be easy to integrate.