Line integral with respect to x or y

[ehrenfest]

I am confused about how

$$\int_C f(x,y) dx = \lim_{||P|| \to 0} \sum_{i = 1}^n f(x_i^*,y_j^*) \Delta x_i$$ is different from $$\int f(x,y) dx$$

where P is a partition and its norm is the length of its largest elements. The index i represents an element in that partition and the asterik means the endpoint closest to the origin of that part of the partition.

 

[quasar987]

Is there a context? Where did you see that they were different things? The classical notation for a line integral in the plane is

$$\int_C P(x,y)dx+Q(x,y)dy$$

where this is to be understood as

$$\int_{a}^{b}P(x(t),y(t))\frac{dx}{dt}dt+\int_{a}^{b}Q(x(t),y(t))\frac{dy}{dt}dt$$

Where t is the parameter for the curve C.

But if there is no Q(x,y), the integral turns out to be equivalent to a simple integration over x like you wrote.

 

[ehrenfest]

Is there a context? Where did you see that they were different things? The classical notation for a line integral in the plane is

$$\int_C P(x,y)dx+Q(x,y)dy$$

where this is to be understood as

$$\int_{a}^{b}P(x(t),y(t))\frac{dx}{dt}dt+\int_{a}^{b}Q(x(t),y(t))\frac{dy}{dt}dt$$

Where t is the parameter for the curve C.

But if there is no Q(x,y), the integral turns out to be equivalent to a simple integration over x like you wrote.

So, you are saying

$$\int_{a}^{b}P(x(t),y(t))\frac{dx}{dt}dt = \int_{x(a)}^{x(b)}P(x,y)dx$$

That seems unintuitive to me for some reason.

 

[quasar987]

It's just the change of variable formula!