Can someone simply explain to me how to calculate linear integral linke below?
$$\int_{L} 5y \mathrm{d}L$$
Where L is line segment from (0;0) to (0,2;0,2).
Can someone simply explain to me how to calculate linear integral linke below?
$$\int_{L} 5y \mathrm{d}L$$
Where L is line segment from (0;0) to (0,2;0,2).
Assuming you mean to calculate the line integral $\int_L {5y \, {\rm d}s}$, where $L$ is the segment from $(0,0)$ to $(0.2,0.2)$, then define a bijective parametrization $r:[0,0.2] \to L$ by $r(t)=(t,t)$. Also let $f(x,y)=5y$. Then, $$ \int_L {f\,{\rm d}s} = \int_0^{0.2} {f(r(t))|r'(t)|\,{\rm d}t}. $$ Here, $f(r(t)) = f(t,t) = 5t$ and $|r'(t)| = |(1,1)| = \sqrt{2}$. Thus, $\int_L {5y \, {\rm d}s} = \sqrt 2 \int_0^{0.2} {5t \, {\rm d}t} $.