How do you solve the following where $k \in \mathbb{N}$? \begin{align*} g(k) := \int_{B(0, 1) \subseteq \mathbb{R}^d} 1 - |u|^k du \end{align*} For example, we have $g(1) = \frac{v_d}{d+1}$ and $g(2) = \frac{2v_d}{d+2}$ where $v_d$ is the volume of $B(0, 1)$, the unit ball in $\mathbb{R}^d$. How about for general $k$?
Integrating (1 - |u|^k) over unit ball in R^d
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integration
1 Answers
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The surface area of the unit ball in $\mathbb{R}^d$ is $S_d = 2\pi^{d/2}/\Gamma(d/2)$, and the volume is $S_d/d$. The integral does not depend on the angular variables, so $$ \int_{B(0,1)} (1-\lvert u \rvert^k) \, du = S_d \int_0^1 (1-r^k)r^{d-1} \, dr = S_d \left( \frac{1}{d} - \frac{1}{d+k} \right) = \frac{S_d}{d} \frac{k}{d+k} = v_d \frac{k}{d+k}. $$