The approach of an area calculation discussed before generalises to the calculation of the area of a region enclosed by the \(x\)-axis, two vertical lines with equations \(x=a\) and \(x=b\), and the graph of a function \(f(x).\) Let \(F(x)\) be a function such that \(F'(x)=f(x)\), then \(F(x)\) is called a primitive function or antiderivative of \(f(x).\) Such a primitive function is not uniquely determined: for any constant \(c\), the function \(G(x)=F(x)+c\) is also a primitive function of \(f(x)\).

Let \(O(x)\) be the area enclosed by the \(x\)-axis, the vertical lines through the points \((a,0)\) and \((x,0)\), and the graph of \(f(x)\). The figure below illustrates this situation.

Just as in the concrete example discussed earlier one can look at the difference \(O(x+\dd x)-O(x)\), that is, the area of the region enclosed by the \(x\)-axis, the vertical lines through the points \((x,0)\) and \((x+\dd x,0)\), and the graph of \(f(x)\), for a very small increase \(\dd x.\) As before, the additional narrow strip can be approximated for very small \(\dd x\) by a rectangular strip of width \(\dd x\) and height \(f(x)\). This is actually for infinitesimal \(\dd x\) nothing else than the differential of the function \(O(x)\). Thus: \[\dd O=f(x)\,\dd x.\] In other words, \[O\,'(x)=f(x).\] So the function \(O(x)\) is a primitive function of \(f(x)\) and, up to an additional constant, equal to \(F(x).\) Because \(O(a)=0\) the following equality is valid: \(O(x)=F(x)-F(a).\) In particular, it we have \(O(b)=F(b)-F(a).\) This number \(F(b)-F(a)\) is called the definite integral of the function \(f(x)\) over the closed interval \([a,b]\) and one uses the following format: \[\int_a^b f(x)\,\dd x=F(b)-F(a).\] The function \(f(x)\) is called the integrand and the sign \(\int_{}^{}\) in front of the differential \(f(x)\,\dd x\) is called the integral sign. This sign was invented by Leibniz, one of the pioneers in the field of differential and integral calculus.

Actually, the variable of integration \(x\), which usually designates a position, does not play any particular role in the above calculation, and can be replaced by any other available letter, for example, by the letter \(t\) that is usually reserved for time. The context of the calculation of the integral changes, but the calculation itself remains essentially the same.

Instead of \(F(b)-F(a)\) one often writes \(\bigl[F(x)\bigr]^b_a\) or \(\bigl[F(x)\bigr]^{x=b}_{x=a}\). So:
\[\int_a^bf(x)\,\dd x=\bigl[F(x)\bigr]^b_a = F(b)-F(a)\text.\]