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 ?

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