Proof of The Corollary
Suppose F is an antiderivative of f, with f continuous on . Let
- .
By the first part of the theorem, we know G is also an antiderivative of f. It follows by the mean value theorem that there is a number c such that G(x) = F(x) + c, for all x in . Letting x = a, we have
which means c = − F(a). In other words G(x) = F(x) − F(a), and so
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“Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.”
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