- #1
vanmaiden
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Homework Statement
I am in the process of studying integration and finding the areas under curves. So far, I know of two methods of finding the area under a curve: the limit method and the direct integral method. Could someone explain the relationship between these two methods?
Homework Equations
[itex]\int[/itex]f(x) dx = F(x)|[itex]^{b}_{a}[/itex] = F(b) - F(a) = Area
[itex]lim_{n→∞}[/itex] [itex]\sum^{n}_{i = 1}[/itex] [itex]f(x_{i})[/itex]Δx = Area
The Attempt at a Solution
I noticed in the direct integration method for finding the area under a curve that the area under the curve is equal to the change in y of a more complicated function: the integral. I graphed it out on my calculator and I don't see exactly how this works.
[itex]lim_{n→∞}[/itex] [itex]\sum^{n}_{i = 1}[/itex] [itex]f(x_{i})[/itex]Δx = Δy of F(x) = Area
I'm trying to seek an explanation as to why the limit method yields the same result as the direct integral method.