Expert answer:Proving the Hypervolume of a Hypersphere using a Q

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Solved by verified expert:The assignment is to prove the hypervolume of a hypersphere with a quadruple integral using trigonometric substitutions and integral formulas (given) to confirm/prove the the hypervolume of a hypersphere. The hypervolume equation is given as ((pi^2)(a^4))/2. The assignment has more detailed information and reference material.We did a similar activity in class, but as I have never attempted a quadruple integral I wanted to be able to have a solution to compare my work to for the hypersphere. Thanks!
quadruple_integrals.pdf

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Failure to do so can be considered a violation of Colgate’s Academic Honor Code.
Question:
Use a quadruple integral, trigonometric substitution(s), and integral formulas for sinn (u) and
cosn (u) found inside the back cover of our text to confirm that
π 2 a4
gives the hypervolume of a
2
hypersphere of radius a: x 2 + y 2 + z 2 + w 2 = a2
Other Information:
This assignment is an extension of an activity that we did in class. The completed activity
from class is attached below:
Continue your work on the reverse
The formulas given from the book are:
He stressed the importance of formula #17 and 18.
The question itself is basically a proof, I’m just a little confused as to how to solve the
problem/the work involved as we have never done a quadruple integral before. I will also solve
the problem myself to the best of my ability based off of what we did in class, but I just wanted
to be sure that I do everything correctly in the end and have the ability to check my work.
Thanks!
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