The Volume and the Surface Area of a Ellipsoidal CAP 

Author  HouLinBo 
Tutor  ZhouJiaZu 
School  Guizhou Normal 
Course  Computational Mathematics 
Keywords  ellipsoidal cap spheral cap area formula volume formula sphere 
CLC  O123.2 
Type  Master's thesis 
Year  2007 
Downloads  494 
Quotes  0 
In this thesis we wish to solve two problems.First we will obtain the volume and the surface area of a ellipsoidal cap.The standard equation of n+1axis ellipsoid in n+1dimensions Euclidean spaceE_{n+1}is where a_{i}＞0,i=1,2,...,n+1.n≥1.Let x_{i},（i=1,2,...,n+1）be equal to h（h∈[0,a_{i}]）on x_{i}axis,x_{i}=h is a hyperplane and is a vertical plane of x_{i}axis.Since the part of ellipsoidal surface between hyperplane x_{i}=h and x_{i}=a_{i} is said to be the ellipsoidal cap of n+1axis ellipsoid.Furthermore we will obtain the volume formula of the ellipsoidal cap during hyperplane x_{1}=a_{1} and x_{1}=h.Consider ellipsoidal cap of n+1axis ellipsoid x_{1}^{2}/a_{1}^{2} + x_{2}^{2}/a_{2}^{2} +...+ x_{n+1}^{2}/a_{n+1}^{2}≤1 in E_{n+1},the part between hyperplane x_{1}=a_{1} and x_{1}=h,（h∈[0,a_{1}]）is called ellipsoid cap.Let V_{n+1}（cap）be the volume of the ellipsoid cap.We obtain the volume formula of the ellipsoidal cap where n≥1, I_{0}=π/2arcsinh/a_{i}, I_{1}=1h/a_{1} V_{n+1}=Vol（sum from i=1 to n+1（x_{i}/a_{i}）^{2}≤1）is the volume of n+1axis ellipsoid V_{0}（cap）=1,and V_{1}（cap）=a_{1}h,Γ（.）is the Gamma function. In the third part we get the expression of the surface area of ellipsoid sum from i=1 to n+1 x_{i}^{2}/a_{i}^{2}= 1.Let the standard equation of n+1axis ellipsoid in n+1dimensions Euclidean space be then the area formula of ellipsoidal surface is where