Physlib.SpaceAndTime.Space.Integrals.NormPow

Let $\text{Space } d$ be a $d$-dimensional real inner product space equipped with the Euclidean norm $\|\cdot\|$. For any positive integer $d$, positive radius $b \in \mathbb{R}$, and exponent $p \in \mathbb{R}$, the function $x \mapsto \|x\|^p$ is integrable on the open ball $B(0, b) = \{x \in \text{Space } d : \|x\| < b\}$ if and only if $d + p > 0$.

For a positive natural number $d$ and the $d$-dimensional real inner product space $\text{Space } d$, let $a$ be a positive real number and $p$ be a real exponent. The function $f(x) = \|x\|^p$ is integrable on the complement of the ball centered at the origin with radius $a$, $\{x \in \text{Space } d \mid \|x\| \geq a\}$, if and only if $d + p < 0$.

Let $\text{Space } d$ be a $d$-dimensional real inner product space. For any real number $a > 0$ and any real numbers $b$ and $p$, the function $x \mapsto \|x\|^p$ is integrable on the shell defined by $\{x \in \text{Space } d \mid a \le \|x\| < b\}$.

Let $\text{Space } d$ be a $d$-dimensional real inner product space with dimension $d > 0$. Let $s \subseteq \text{Space } d$ be a bounded set that is also a neighborhood of the origin. For any real exponent $p$, the function $f(x) = \|x\|^p$ is integrable on $s$ if and only if $d + p > 0$, where $\|\cdot\|$ denotes the Euclidean norm on $\text{Space } d$.

Let $\text{Space } d$ be a $d$-dimensional real inner product space. For any bounded subset $s \subseteq \text{Space } d$ such that the origin is in the exterior of $s$ (i.e., the complement $s^c$ is a neighborhood of $0$), and for any real exponent $p$, the function $x \mapsto \|x\|^p$ is integrable on $s$.

Let $d$ be a positive natural number and $\text{Space}(d)$ be a $d$-dimensional real inner product space. Let $s \subseteq \text{Space}(d)$ be a set such that its complement $s^c$ is a neighborhood of the origin (implying the origin is in the exterior of $s$). For any real number $p$, if $d + p < 0$, then the function $x \mapsto \|x\|^p$ is integrable on $s$.