Motivation behind definitions of the Integral without reference to Derivatives If a (definite) Integral can simply be calculated as the difference of two Antiderivatives and Antiderivatives are simply the "reverse process" of Differentiation. Then it seems to me that the Integral can be entirely defined in terms of the Derivative. Given that I don't understand the motivation behind the multiple different definitions of the Integral that seems to go through great pains to define it independently of the concept of Differentiation. Such as the Riemann or Lebesgue definitions. Why go through all this trouble if we can easily define it in terms of Derivatives?
 A: Consider the function $$f:[0,1]\to\Bbb R:x\mapsto\begin{cases}0,&\text{if }x\text{ is rational}\\1,&\text{if }x\text{ is irrational}\;.\end{cases}$$
This function has a Lebesgue integral: if $\lambda$ is Lebesgue measure on $[0,1]$, $\int_{[0,1]}f\,d\lambda=1$. However, there is no function $g:[0,1]\to\Bbb R$ such that $f=g'$. This is a simple example of an integral that can’t be obtained from ordinary derivatives.
From a historical and philosophical point of view, integrals and derivatives are solutions to two apparently unrelated problems. On the face of it there is no obvious reason for the problem of finding tangents and instantaneous rates of change to be closely related to the problem of calculating the areas of complicated regions. It’s true that at the most elementary level the fundamental theorem of calculus connects these two problems in a very nice way, but while this connection provides a nice computational shortcut for calculating some areas, that shortcut doesn’t really illuminate the integral as a solution to the problem of finding areas. Each of these problems has its own history and its own generalizations, and there is something to be learned by considering each in its own right.
A: People calculated areas in ancient Greece long before anyone was thinking about differentiation. That differentiation and integration can be related to each other by the fundamental theorem of calculus is an interesting result, but it desn't give us any intuition about what an integral is supposed to represent. That we can calculate the area under a curve using ideas from differential calculus is an illuminating idea that shouldn't be rendered a triviality by making it a definition.
In more advanced treatments, we often take integrals of functions defined on fairly abstract spaces where differentiation in the calculus sense is not applicable. Contemporary probability is built on the theory of measure and integration. Integrals represent expectations there, and we have to make use of the abstract theory of the Lebesgue integral.
