If you define a function [tex]g(x) = \int_a^x \! f(t) \, \mathrm{d} t[/tex] then from what I currently understand, g(x) gives the value of the area under the curve [tex]y=f(t)[/tex](adsbygoogle = window.adsbygoogle || []).push({});

When you differentiate both sides, [tex]g'(x)[/tex] gives the rate of change of the area underneath [tex]y=f(t)[/tex], however, I don't understand intuitively why f(x) is the rate of change of the area underneath its curve.

Also, when why when you differentiate it (the right hand side), is it [tex]f(x)[/tex] and not [tex]f(t)[/tex]? I can't seem to get my head around the whole respect to x thing - what about t? Why not respect to t? Then if you were to integrate it again, would x become the dummy variable?

The dummy variable is really confusing!

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# Questions regarding the Fundamental Theorem of Calculus

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