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i sort of knew this short proof of the fundamental theorem of calculus since a long while...but never actually saw it anywhere in books or any name associated with it.

i know it must be known...but still ive been curious about it so just wanted to ask.

so here goes my proof...

imagine two curves.one is deravitive and the other its antideravitive.we want the area under the deravitive curve.and the fundamental theorem says that the area under the deravitive curve is the difference between two y values of the antideravitive curve.I simply justify that by saying...the area under the deravitive curve as mentioned in many books is sum of infnite tiny rectangles each of area f(x)dx.now this area of each rectangle is also equal to small change dy in antideravitive curve since dy of antideravitive curve is also instantaneous slope times dx or again f(x)dx.thus on left side of fundamental theorem we are adding up infinite small rectangle areas to get the total area...and on right handside we are calculating the sum of infinite dy s by finding the difference between two y values of antideravitive curve.