Dissertation > Mathematical sciences and chemical > Mathematics > Elementary Mathematics > Elementary Geometry > Solid geometry

# 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
• Abstract
• Contents

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+1-axis ellipsoid in n+1-dimensions Euclidean spaceEn+1is where ai＞0,i=1,2,...,n+1.n≥1.Let xi,（i=1,2,...,n+1）be equal to h（h∈[0,ai]）on xi-axis,xi=h is a hyperplane and is a vertical plane of xi-axis.Since the part of ellipsoidal surface between hyperplane xi=h and xi=ai is said to be the ellipsoidal cap of n+1-axis ellipsoid.Furthermore we will obtain the volume formula of the ellipsoidal cap during hyperplane x1=a1 and x1=h.Consider ellipsoidal cap of n+1-axis ellip-soid x12/a12 + x22/a22 +...+ xn+12/an+12≤1 in En+1,the part between hyperplane x1=a1 and x1=h,（h∈[0,a1]）is called ellipsoid cap.Let Vn+1（cap）be the volume of the ellipsoid cap.We obtain the volume formula of the ellipsoidal cap where n≥1, I0=π/2-arcsinh/ai, I1=1-h/a1 Vn+1=Vol（sum from i=1 to n+1（xi/ai2≤1）is the volume of n+1-axis ellipsoid V0（cap）=1,and V1（cap）=a1-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 xi2/ai2= 1.Let the standard equation of n+1-axis ellipsoid in n+1-dimensions Eu-clidean space be then the area formula of ellipsoidal surface is where

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