How to calculate $\int_{|x| < r} |x|^n dm$? Integral over ball.

Question

How to calculate $\int_{|x| < r} |x|^n dm$? For some $r$, say $r=3$.

There are some simpler integrals here:

Indefinite Integral of Absolute Value of x? Is there a closed form solution?

But this one confuses me a bit, since it has the ball and $n$ which may be even or odd.

And here's a particular answer with this in $\mathbb{R}^n$ case, but it lacks details:

Computing the Lebesgue integral over a ball

score 1 · Answer 1 · answered Nov 24 '19 at 15:56

1

\begin{align*} \int_{|x|<r}|x|^{n}dx=\int_{S^{n-1}}\int_{0}^{r}|\rho\theta|^{n}\rho^{n-1}d\rho d\theta=\omega_{n-1}\int_{0}^{r}\rho^{n}\rho^{n-1}d\rho=\omega_{n-1}\cdot\dfrac{1}{2n}\cdot r^{2n}, \end{align*} where $\omega_{n-1}$ is the surface area of the unit sphere in $\mathbb{R}^{n}$.

answered Nov 24 '19 at 15:56

user284331

55,591

Can you explain more, what happens here? – mavavilj Nov 24 '19 at 16:07
The spherical integration method is such that letting $x=\rho\theta$, where $\rho$ runs in the radius of the ball, and $\theta$ runs in $S^{n-1}$. – user284331 Nov 24 '19 at 16:09
Where does $p^{n-1}$ come from? – mavavilj Nov 24 '19 at 16:24
That is the some sort of the result of Jacobian. – user284331 Nov 24 '19 at 16:29
I cannot find a reference that gives the formulation that you give. – mavavilj Nov 24 '19 at 16:37
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Look at Folland Real Analysis Second Edition page 78. – user284331 Nov 24 '19 at 16:40

How to calculate $\int_{|x| < r} |x|^n dm$? Integral over ball.

1 Answers1