Jaime Koshy 

MAT 267 

Naala Brewer 

Volumes of a Hypershere 

 

The principal goal of this investigation was to find a process to calculate the volume enclosed in a hypersphere in the 4^th dimension, real^4.  The secondary goal of this investigation was to find a process to calculate relative volumes enclosed in hyperspheres in the n^th dimension..  To do this, I have to initially start out fundamentally small by finding the area of an enclosed circle in a 2-dimensional space, real^2.  From there, I will progress into the next dimension. 

 

 

So to begin, I used a double integral to find the area of a circle. I used polar coordinates instead of rectangular coordinates for ease of integration. 

 

 

 

 

(`+`(2*Pi, 0))(`+`(1/2*R^2, 0)) = 2*`À`(1/2*R^2) and 2*`À`(1/2*R^2) = `ÀR`^2 

 

 

Second, I used a triple integral to find the volume of a sphere (3-dimensions), real^3.  Here again I used polar coordinates for ease.  The bounds for the third integral is found by solving for z in the equation for a sphere x^2+y^2+z^2 = R^2. 

 

 

 

 

 

 

 

=  

 

 

    I used a  u-du substitution to solve the remaining integral.  

 

=  

 

 

 

 

The next step was to use a quadruple integral to find the hypervolume of a hypersphere, meaning a sphere in four-dimensions, real^4.  Again as in my first two calculations, I used spherical coordinates to avoid difficult integrals involved with rectangular coordinates.  The bounds for the fourth integral is found by solving for w in the equation for a hypersphere x^2+y^2+z^2+w^2 = R^2 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     

 

    To solve the last akward integral, I referenced a table of proved integrals to find:  

 

 

 

 

    Therefore, using the substitutions and my final calculation looked like:  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Principally, my investigation shows that you can keep finding an object's relative volume in the next dimension by augmenting the previous dimension's calculation, which basically translates to adding another integral.  The bounds for the new integral would be the top function minus the bottom function of the object's projection on the previous dimension.  A general formula of this is shown below.  Just because I converted my coordinates does not mean it is impossible with rectangular coordinates.  I opted for the conversion in order to allay arduous integrals.  If using rectangular coordinates, numerous trigonometric substitutions along with references to an integral table will be necessary.  Deriving this formula for rectangular coordinates might be good for a future project using what I have proven in this project to build upon. 

 

 

 

not real^(n-1) 

 

  Solve for n^thvariablewhere the arrow is in the equation for a hypersphere of radius R, x[1]x[2]x[n].  The outer real^(n-1) integrals will be the same as in the previous dimension, real^(n-1). 

 

 

 

*References:  Ms. Naala Brewer and Table of Integrals in Stewart's Essential Calculus Textbook