proof of fundamental theorem of calculus

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hi i have problem understanding the proof of fundamental theorem of calculus. this is part of general proof process using extreme value.

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briefly this above saying that the derivate of integral is getting closed to f(x). But what i don't understand is that the unit of result of integral is area, so as h approaches 0, the integral result should be Zero ((f(x+h)-f(x))*0 = 0), not f(x) as in proof.

Hope somebody help me for this.