(Solved): Hyperspheres An "n-sphere" is the set of points in \( n \)-dimensional Euclidean space that are at ...

Hyperspheres An "n-sphere" is the set of points in \( n \)-dimensional Euclidean space that are at a fixed distance \( a \) from the origin. You know that the volume enclosed by a 3-sphere (the ordinary sphere) is \( 4 \pi a^{3} / 3 \), and the "volume" enclosed by a 2-sphere (the ordinary circle) is \( \pi a^{2} \). Calculate the volume of the 4 -sphere by evaluating the integral \[ V_{4}=\int_{-a}^{a} \int_{-\sqrt{a^{2}-x^{2}}}^{\sqrt{a^{2}-x^{2}}} \int_{-\sqrt{a^{2}-x^{2}-y^{2}}}^{\sqrt{a^{2}-x^{2}-y^{2}}} \int_{-\sqrt{a^{2}-x^{2}-y^{2}-z^{2}}}^{\sqrt{a^{2}-x^{2}-y^{2}-z^{2}}} d u d z d y d x \]