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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

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