We use integration to measure lengths, areas, or volumes. This is a geometrical interpretation, but we want to examine an analytical interpretation that leads us to Integration reverses differentiation. Hence let us start with differentiation.
Weierstraß Definition of Derivatives
##f## is differentiable at ##x## if there is a linear map ##D_{x}f##, such that
\begin{equation*}
\underbrace{D_{x}(f)}_{\text{Derivative}}\cdot \underbrace{v}_{\text{Direction}}=\left(\left. \dfrac{df(t)}{dt}\right|_{t=x}\right)\cdot v=\underbrace{f(x+v)}_{\text{location plus change}}-\underbrace{f(x)}_{\text{location}}-\underbrace{o(v)}_{\text{error}}
\end{equation*}
where the error ##o(v)## increases slower than linear (cp. Landau symbol). The derivative can be the Jacobi-matrix, a gradient, or simply a slope. It is always an array of numbers. If we speak of derivatives as functions, then we mean ##f'\, : \,x\longmapsto D_{x}f.## Integration is the problem to compute ##f## from ##f'## or ##f## from
$$
\dfrac{f(x+v)-f(x)}{|v|}+o(1)
$$
The quotient in this expression is linear in ##f## so
$$
D_x(\alpha f+\beta g)=\alpha D_x(f)+\beta D_x(g)
$$
If we add the Leibniz rule
$$
D_x(f\cdot g)=D_{x}(f)\cdot g(x) + f(x)\cdot D_{x}(g)
$$
and the chain rule
$$
D_x(f\circ g)=D_{g(x)}(f)\circ g \cdot D_x(g)
$$
then we have the main properties of differentiation.