- #1

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Hi,

I have been trying to understand the meaning of line integrals, as opposed to the good ol' fashioned regular integral. Graphically, an integral like :

[tex] \int_{a}^{b} f(x) dx = \lim_{\|\triangle\|\rightarrow 0} \sum_{i} f(w

_i) \triangle x_i [/tex]

tells me that the area between the x-axis and the graph, from x=a to x=b is my integral. This was done by dividing that area into rectangles with width delta Xi and length f(Wi)

now how about the line integral :

[tex] \int_{C} f(x,y) ds = \lim_{\|\triangle\|\rightarrow 0} \sum_{i} f(u_i, v_i) \triangle s_i [/tex]

Now compared to my understanding of the definite integral, what does this definition exactly mean?

1. What is f(ui, vi) ?

2. "del si" is the arc length on our curve, but what does multiplying the arc length by f(ui, vi) give us?

I have been trying to understand the meaning of line integrals, as opposed to the good ol' fashioned regular integral. Graphically, an integral like :

[tex] \int_{a}^{b} f(x) dx = \lim_{\|\triangle\|\rightarrow 0} \sum_{i} f(w

_i) \triangle x_i [/tex]

tells me that the area between the x-axis and the graph, from x=a to x=b is my integral. This was done by dividing that area into rectangles with width delta Xi and length f(Wi)

now how about the line integral :

[tex] \int_{C} f(x,y) ds = \lim_{\|\triangle\|\rightarrow 0} \sum_{i} f(u_i, v_i) \triangle s_i [/tex]

Now compared to my understanding of the definite integral, what does this definition exactly mean?

1. What is f(ui, vi) ?

2. "del si" is the arc length on our curve, but what does multiplying the arc length by f(ui, vi) give us?

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