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Why do we integrate a function to find the area under it? 
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#1
Jan2410, 05:37 AM

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Why, when finding the area by definite integral, we have to find the indefinite integral first? As I understand, to find the area of under the curve, all we need is the equation of the curve. On the other hand, the indefinite integral helps us to find the original function from its derivative. So what does this have to do with finding the area?



#2
Jan2410, 05:51 AM

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That is the truly beautiful insight in the fundamental theorem of calculus: To sum up the area beneath some curve, essentially an INFINITE process, can trivially be done by finding an antiderivative to the defining curve. 


#3
Jan2410, 07:55 AM

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#4
Jan2410, 08:08 AM

P: 240

Why do we integrate a function to find the area under it?
Which helps us in calculating integrals more efficiently. 


#5
Jan2410, 08:46 AM

P: 87

But it is true that the area under a curve is equal to the area under that curve's derivative, right?



#6
Jan2410, 09:59 AM

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#7
Jan2410, 11:01 AM

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#8
Jan2410, 01:18 PM

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#9
Jan2410, 01:28 PM

P: 256

Juwane: That quote you gave was just a restatement of the Fundamental Theorem of Calculus, nicksauce already gave an excellent example of where your assertion is false.



#10
Jan2410, 07:06 PM

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#11
Jan2510, 06:39 PM

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@Juwane: how do get your assertion from the wikipedia quote? If [tex]A_f(x)[/tex] denotes the area function of some function f(x), then:
wiki (i.e. the fundamental theorem) says [tex]A_f'(x)=f(x)[/tex]; you're saying that [tex]A_f(x)=A_{f'}(x)[/tex]. 


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