Definite Integral of Step Function

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Theorem

Let $\alpha, \beta \in \R$ be a real numbers such that $\alpha < \beta$.

Let $f \left({x}\right)$ be a step function defined on the interval $\left[{\alpha \,.\,.\, \beta}\right]$:

$f \left({x}\right) = \lambda_1 \chi_{\mathbb I_1} + \lambda_2 \chi_{\mathbb I_2} + \cdots + \lambda_n \chi_{\mathbb I_n}$

where:

$\lambda_1, \lambda_2, \ldots, \lambda_n$ are real constants
$\mathbb I_1, \mathbb I_2, \ldots, \mathbb I_n$ are intervals, where these intervals partition $\left[{\alpha \,.\,.\, \beta}\right]$
$\chi_{\mathbb I_1}, \chi_{\mathbb I_2}, \ldots, \chi_{\mathbb I_n}$ are characteristic functions of $\mathbb I_1, \mathbb I_2, \ldots, \mathbb I_n$.


Then the definite integral of $f$ {with respect to $x$ over $\left[{\alpha \,.\,.\, \beta}\right]$ is given by:

$\displaystyle \int_\alpha^\beta f \left({x}\right) \, \mathrm d x = \sum_{k \mathop = 1}^n \lambda_k \left({\beta_k - \alpha_k}\right)$

where $\alpha_k, \beta_k$ are the endpoints of $\mathbb I_k$ for $1 \le k \le n$.


Proof

Each of the intervals $\mathbb I_k$ is such that $f \left[{\mathbb I_k}\right]$ is a constant function:

$\forall x \in \mathbb I_k: f \left({x}\right) = \lambda_k$

Thus:

\(\displaystyle \int_{\mathbb I_k} f \left({x}\right) \, \mathrm d x\) \(=\) \(\displaystyle \int_{\alpha_k}^{\beta_k} \lambda_k \, \mathrm d x\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \lambda_k \left({\beta_k - \alpha_k}\right)\) $\quad$ Definite Integral of Constant $\quad$


From the corollary to Sum of Integrals on Adjacent Intervals for Integrable Functions:

\(\displaystyle \int_\alpha^\beta f \left({x}\right) \, \mathrm d x\) \(=\) \(\displaystyle \sum_{k \mathop = 1}^n \int_{\mathbb I_k} f \left({x}\right) \, \mathrm d x\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 1}^n \lambda_k \left({\beta_k - \alpha_k}\right)\) $\quad$ $\quad$

$\blacksquare$