I know from vector calculus, that line integral is the area of the curtain under the curve. Then, i'm realize we can solve the line integral with respect to $x$ or $y$ .
Integration with respect to $y$ is just like ordinary definite integral. But what about with respect to $x$? Is it an arc length?
And i'm still confuse with the closed integral. Is this kind of integral calculate the area which is bounded by closed path?
Please give me the best explanation. Thanks. Anyway, my teacher just teached me about calculating, not a concept. That's why i don't understand a bit.
You have two kinds of line integrals.
One where $F(x,y) = z$ gives has a one dimensional output, and represents a surface in $\mathbb R^3$. You are integrating over some path $\mathbb r$ in the $xy$ plane that is parametrized by $t.$
$\int_r F(x,y)\ d\mathbb r = \int F(x,y) \|\mathbb r(t)\|\ dt$
One way to visualize this, is that you have some curtain that falls from the surface directly above your path down to the path, and what is the area of that curtain.
The other $F(x,y) = (P(x,y),Q(x,y))$ gives a multi-dimensional, or vector as an output. Now we traditionally think of $F$ as representing a force-field.
$\int_r F(x,y)\cdot d\mathbb r$ represents the amount of energy that will be required to move though this force-field.