Proof of the First Fundamental Theorem of Calculus The ﬁrst fundamental theorem says that the integral of the derivative is the function; or, more precisely, that it’s the diﬀerence between two outputs of that function. Theorem: (First Fundamental Theorem of Calculus) If f is continuous and �b F � = f , then a f (x) dx = F (b) − F (a). Proof: By using Riemann sums, we will deﬁne an antiderivative G of f and then use G(x) to calculate F (b) − F (a). We start with the fact that F � = f and f is continuous. (It’s not strictly necessary for f to be continuous, but without this assumption we can’t use the second fundamental theorem in � xour proof.) Next, we deﬁne G(x) = a f (t) dt. (We know that this function exists because we can deﬁne it using Riemann sums.) The second fundamental theorem of calculus tells us that: G� (x) = f (x) So F � (x) = G� (x). Therefore, (F − G)� = F � − G� = f − f = 0 Earlier, we used the mean value theorem to show that if two functions have the same derivative then they diﬀer only by a constant, so F − G = constant or F (x) = G(x) + c. This is an essential step in an essential proof; all of calculus is founded on the fact that if two functions have the same derivative, they diﬀer by a constant. Now we compute F (b) − F (a) to see that it is equal to the deﬁnite integral: F (b) − F (a) = (G(b) + c) − (G(a) + c) = G(b) − G(a) � a � b = f (t) dt − f (t) dt a � a b f (t) dt − 0 = a � F (b) − F (a) = b f (x) dx a