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Finding the Relationship between the Limit Method and Direct Integral Method for Area 
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#1
Nov3011, 09:36 PM

P: 101

1. The problem statement, all variables and given/known data
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? 2. Relevant 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 3. 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. 


#2
Nov3011, 10:13 PM

HW Helper
Thanks
PF Gold
P: 7,659

That is the fundamental theorem of calculus. You might start by reading here:
http://en.wikipedia.org/wiki/Fundame...em_of_calculus 


#3
Dec311, 03:08 AM

P: 101

F(x) = [itex]\int^{x}_{a}f(t) dt[/itex] I've just never seen an antiderivative represented in this way before. Could you interpret this for me? Why does an antiderivative have an upper and lower bound? 


#4
Dec311, 12:07 PM

HW Helper
Thanks
PF Gold
P: 7,659

Finding the Relationship between the Limit Method and Direct Integral Method for Area
Now the x in that definite integral is a dummy variable, not affecting the answer, so that line could as well have been written[tex]\int_a^b f(t)\,dt = F(t)_a^b = F(b)  F(a)[/tex] Since this is true for any a and b, let's choose to let b be a variable x:[tex]\int_a^x f(t)\,dt = F(t)_a^x = F(x)  F(a)[/tex] Since these are equal you still have F'(x) = f(x) so the left side is an antiderivative of f(x). Since a can be anything, the F(a) is like the constant of integration in our first equation. Does that help answer your question? 


#5
Dec311, 12:16 PM

P: 101




#6
Dec311, 12:32 PM

HW Helper
Thanks
PF Gold
P: 7,659




#7
Dec311, 01:00 PM

P: 101

If f(0) = 4, then shouldn't f(x)  f(0) = [itex]\frac{x^{3}}{3}[/itex]  4?



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