Volume of a unit hyper sphere of radius 1

Inspired by foundations of data science, the area of a circle is

\( \int_{x=-1}^{x=1}\int_{y=-\sqrt{1-x^2}}^{y=\sqrt{1-x^2}}dydx \)

This further extended to a sphere is

\(\ int_{x=-1}^{x=1}\int_{y=-\sqrt{1-x^2}}^{y=\sqrt{1-x^2}}\int_{z=-\sqrt{1-x^2-y^2}}^{z=\sqrt{1-x^2-y^2}}dzdydx \)

This when implemented via maxima is

(%i17) integrate(integrate(integrate(1, z, -sqrt(1-x^2-y^2), sqrt(1-x^2-y^2)),

        y, -sqrt(1-x^2), sqrt(1-x^2)), x, -1, 1);

Maxima goes on to ask if

"Is "(x-1)*(x+1)" positive or negative?"

For a circle this value is definitely negative and voila we get the answer as $4\pi/3$. Higher dimenisons lead to interesting results

For 4 dimensions, we use

integrate(integrate(integrate(integrate(1, x4, -sqrt(1-x1^2-x2^2-x3^2), 

          sqrt(1-x1^2-x2^2-x3^2)), x3, -sqrt(1-x1^2-x2^2), 

          sqrt(1-x1^2-x2^2)), x2, -sqrt(1-x1^2), sqrt(1-x1^2)), 

          x1, -1, 1);

and get the volume as $\pi^2/2$ as the answer which matches what the book predicts