Definite Integral

So far we have dealt with indefinite integrals — families of functions differing by a constant $C$. A definite integral, written $\displaystyle\int_a^b f(x)\,dx$, is different: it has one fixed numerical value, determined by a lower limit $a$ and an upper limit $b$.

One way to make sense of this value is through the area function. If $f$ is continuous on $[a,b]$, define $A(x) = \displaystyle\int_a^x f(t)\,dt$ as the area of the region bounded by the curve $y=f(x)$, the ordinates at $a$ and at the variable point $x$, and the $x$-axis. This connects directly back to differentiation:

In other words, the rate at which the area function grows is exactly the height of the curve at that point — differentiating the area function returns the original function. This theorem is what links the geometric idea of "area under a curve" back to anti-derivatives, and leads to the practical rule actually used to evaluate definite integrals: