Multiple Dimensions - Volume Of a 4-Sphere

Calculus Level 3

Let's consider how to find the volume of the sphere with radius \(r\) in 4 dimensions. The sphere is the set of points such that \[ x_1 ^2 + x_2 ^2 + x_3 ^2 + x_ 4 ^2 \leq r^2. \]

Let's calculate the volume by integrating along the \( x_1 \) axis. The volume is equal to \( V(r) = \int_{-r} ^{r} A(x_1) \, d x_1 \), where \( A(x_1) \) denotes the area element for a given value of \( x_1\).

What is the area element \( A(x) \) and what is the corresponding volume \( V(r) \) of the sphere ?

Image credit: Wikipedia Pbroks13
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