I am using this baby Rudin definition for Riemann integral:

Let $[a, b]$ be a given interval. By a partition $P$ of $[a, b]$ we mean a finite set of points $x_0, x_1, \ldots, x_n$, where $$ a = x_0 \leq x_1 \leq \cdots \leq x_{n-1} \leq x_n = b.$$ We write $$ \Delta x_i = x_i - x_{i-1} \qquad (i = 1, \ldots, n). $$ Now suppose $f$ is a bounded real function defined on $[a, b]$. Corresponding to each partition $P$ of $[a, b]$ we put $$ \begin{align} M_i &= \sup f(x) \qquad (x_{i-1} \leq x \leq x_i), \\ m_i &= \inf f(x) \qquad (x_{i-1} \leq x \leq x_i), \\ U(P, f) &= \sum_{i=1}^n M_i \Delta x_i, \\ L(P, f) &= \sum_{i=1}^n m_i \Delta x_i, \end{align} $$ and finally $$ \begin{align} \tag{1} \overline{\int_a^b} f dx &= \inf U(P, f), \\ \tag{2} \underline{\int_a^b} f dx &= \sup L(P, f),\\\, \end{align} $$ where the $\inf$ and the $\sup$ are taken over all partitions $P$ of $[a, b]$. The left members of (1) and (2) are called the upper and lower Riemann integrals of $f$ over $[a, b]$, respectively.

If the upper and lower integrals are equal, we say that $f$ is Riemann-integrable on $[a, b]$, $$ \tag{3} \int_a^b f dx. $$

How do we get property $\displaystyle\int_a^b f dx=-\displaystyle\int_b^a f dx$ from this definition?

(Code borrowed from here)

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    $\begingroup$ You don't. That's the convention for defining the integral when $a>b$. $\endgroup$ – Lord Shark the Unknown Dec 29 '18 at 11:19

You can't, because the definition you quoted deals with integrals for $a>b$. You have to further define that $\int_a^afdx=0$ and $\int_b^afdx=-\int_a^bfdx$.

Why you ask? Because it's logical. Negative sign is often asigned to changing the orientation. Also, it plays nicely with the "area" interpretation of integrals.

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    $\begingroup$ Yeah. And also, integration by substitution becomes more applicable. Rudin gives theorem only for $\varphi$ monotonically increasing, but Wikipedia says that it is true for any differentiable $\varphi$. $\endgroup$ – Silent Dec 29 '18 at 11:34
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    $\begingroup$ @Silent Yup. You can sort of break the interval into pieces where the function is monotonic and change the signs if you need to make it monotonically increasing to use that definition given in Rudin. $\endgroup$ – SinTan1729 Dec 29 '18 at 11:49

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