User:MontanaMako/sandbox1

In calculus, the King's rule, also known as the King rule, King property, or King's property, is a property that states that the integral from $a$ to $b$ of a function $f(x)$ with respect to $x$ is equal to the integral from $a$ to $b$ of $f(a+b-x)$ with respect to $x$. In other terms:

$\int \limits _{a}^{b}f(x)dx=\int \limits _{a}^{b}f(a+b-x)dx$

In addition to this, the King's rule states that if a function $f(x)$ defined on the interval $[a,b]$ equals $f(a+b-x)$, then the integral from $a$ to $b$ of $f(x)$ with respect to $x$ is equal to two times the integral from $a$ to ${\frac {a+b}{2}}$ of $f(x)$ with respect to $x$. In other terms, if a function $f(x)$ is symmetric about the midpoint of the interval $[a,b]$, then:

$\int \limits _{a}^{b}f(x)dx=2\int \limits _{a}^{\frac {a+b}{2}}f(a+b-x)dx$