I taught line integration by myself. My textbook provide me two definition of line integration to integrate vector field.
- $\int^a_bf(x(t),y(t))\sqrt{(x'^2(t)+y'^2(t)}dt$
- $\int^a_b \vec F(\vec r(t))\cdot \vec r'(t)dt$
I try to use the first definition to calculate a problem from my textbook
F(x,y) = $xy(i)+3y^2(j)$, r(t) = $11t^4(i)+t^3(j)$ ,$0<t<1$
my calculation: I consider x as $11t^4$, y as $t^3$ and then get $\int^1_0 (11t^3)(t^3)+3(t^3)\sqrt{44t^3+3t^2}dt$ however, the answer is wrong. Can anyone tell me what's the wrong with my calculation?