Volume of a Hypersphere #dimensions #4d #tibees #math

Okay, listen up. We can use calculus to find the volume of the sphere. What we do is to split the sphere up into tiny cylinders. The volume of each cylinder is equal to pi r squared, which is the face area times the width, the x. With the widths getting infinitely small, we integrate or add them all up. To get four thirds pi r cubed has our sphere's volume. We can do the same thing to get the volume of a four-dimensional hypersphere, but this time we add up infinitely many tiny four-d cylinders. They have spheres for faces and are separated by a distance dx. Integrate those, it's a messy integral, but we get pi squared over two times r to the four as our volume of a four-d sphere with radius r.