- #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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