1.7: The Fundamental Theorem of Calculus

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Using the definition of the derivative, we differentiate the following integral: \[\begin{aligned} \frac{d}{dx}\int_a^x f(s)ds&=\underset{h\to 0}{\lim}\frac{\int_a^{x+h} f(s)ds-\int_a^x f(s)ds}{h} \\ &=\underset{h\to 0}{\lim}\frac{\int_x^{x+h} f(s)ds}{h} \\ &=\underset{h\to 0}{\lim}\frac{hf(x)}{h} \\ &=f(x).\end{aligned}\]

This result is called the fundamental theorem of calculus, and provides a connection between differentiation and integration.

The fundamental theorem teaches us how to integrate functions. Let \(F(x)\) be a function such that \(F' (x) = f(x)\). We say that \(F(x)\) is an antiderivative of \(f(x)\). Then from the fundamental theorem and the fact that the derivative of a constant equals zero, \[F(x)=\int_a^x f(s)ds+c.\nonumber\]

Now, \(F(a) = c\) and \(F(b) =\int_a^b f(s)ds + F(a)\). Therefore, the fundamental theorem shows us how to integrate a function \(f(x)\) provided we can find its antiderivative:

\[\label{eq:1}\int_a^b f(s)ds=F(b)-F(a).\]

Unfortunately, finding antiderivatives is much harder than finding derivatives, and indeed, most complicated functions cannot be integrated analytically.

We can also derive the very important result \(\eqref{eq:1}\) directly from the definition of the derivative (1.3.1) and the definite integral (1.6.1). We will see it is convenient to choose the same h in both limits. With \(F'(x) = f(x)\), we have \[\begin{aligned} \int_a^b f(s)ds&=\int_a^b F'(s)ds \\ &=\underset{h\to 0}{\lim}\sum\limits_{n=1}^N F'(a+(n-1)h)\cdot h \\ &=\underset{h\to 0}{\lim}\sum\limits_{n=1}^N \frac{F(a+nh)-F(a+(n-1)h)}{h}\cdot h \\ &=\underset{h\to 0}{\lim}\sum\limits_{n=1}^N F(a+nh)-F(a+(n-1)h).\end{aligned}\]

The last expression has an interesting structure. All the values of \(F(x)\) evaluated at the points lying between the endpoints \(a\) and \(b\) cancel each other in consecutive terms. Only the value \(−F(a)\) survives when \(n = 1\), and the value \(+F(b)\) when \(n = N\), yielding again \(\eqref{eq:1}\).