What is the purpose of the fundamental theorem of calculus?

May 30, 2015

The fundamental theorem of calculus is a simple theorem that has a very intimidating name. It is essential, though. So, don't let words get in your way. This theorem gives the integral the importance it has.

The fundamental theorem of calculus has two parts. The first one is the most important: it talks about the relationship between the derivative and the integral.The first fundamental theorem of calculus states that, if f is continuous on the closed interval $\left[a , b\right]$ and $F$ is the indefinite integral of $f$ on $\left[a , b\right]$, then

${\int}_{a}^{b} f \left(x\right) \mathrm{dx} = F \left(b\right) - F \left(a\right) .$

The second fundamental theorem of calculus holds for f a continuous function on an open interval I and a any point in I, and states that if F is defined by

$F \left(x\right) = {\int}_{a}^{x} f \left(t\right) \mathrm{dt}$,

then

${F}^{'} \left(x\right) = f \left(x\right)$

at each point in $I$.

The fundamental theorem of calculus along curves states that if $f \left(z\right)$ has a continuous indefinite integral $F \left(z\right)$ in a region $R$ containing a parameterized curve $\gamma : z = z \left(t\right)$ for $\alpha \le t \le \beta$, then

${\int}_{\gamma} f \left(z\right) \mathrm{dz} = F \left(z \left(\beta\right)\right) - F \left(z \left(\alpha\right)\right)$