Let $B$ be a unit ball in $\mathbb{R}^3$. Find the average distance from points in $B$ to the origin.
My progress: If I am not mistaken, this problem should be solved by triple integrals and the function should be $x^2+y^2+z^2$ with division by area of unit ball. But I am unsure on which constraints should we construct that triple integral? Any help or suggestion would be welcomed!
Best Answer
A simpler approach is integrating over distance from the origin. We can write our integral as $\frac1V\int rdS$, where $dS$ represents the surface area of the hollow sphere of a given radius. In other words, $dS = 4\pi r^2dr$. So, our integral is $$\frac1V\int\limits_0^14\pi r^3dr=3\int\limits_0^1r^3dr=\frac34r^4\bigg\vert_0^1=\frac34$$