The Student Room Group

Why does the Definite Integral work?

Hi, I am an FSMQ student who has just started looking at the definite integral. I am interested in why the definite integral is actually the area under the curve. I watched a useful Youtube video which defines the definite integral like this:



However, I am confused as to how you get from that definition to f'(b) - f'(a).

If you could explain this I would appreciate it, though if the concepts are too advanced then I'd understand.

Thanks for your help.
Original post by Hariex
Hi, I am an FSMQ student who has just started looking at the definite integral. I am interested in why the definite integral is actually the area under the curve. I watched a useful Youtube video which defines the definite integral like this:



However, I am confused as to how you get from that definition to f'(b) - f'(a).

If you could explain this I would appreciate it, though if the concepts are too advanced then I'd understand.

Thanks for your help.


Does this help ?
Reply 2
Original post by brianeverit
Does this help ?


It doesn't seem to specify how it goes from the definition (in OP) to f'(b) - f'(a) - it just seems to assume we know why. Thanks for your help, though.
Original post by Hariex
It doesn't seem to specify how it goes from the definition (in OP) to f'(b) - f'(a) - it just seems to assume we know why. Thanks for your help, though.


This essentially comes from the fundamental theorem of calculus. The proof is a little involved though.
Reply 4
Original post by ThatPerson
This essentially comes from the fundamental theorem of calculus. The proof is a little involved though.


Could you point me to what the proof looks like? I'm not expecting to understand it; I'm just curious.
Original post by Hariex
Could you point me to what the proof looks like? I'm not expecting to understand it; I'm just curious.


https://math.berkeley.edu/~peyam/Math1ASp11/Proof%20of%20the%20FTC.pdf

http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Second_part

Quick Reply

Latest