The Volume and the Surface Area of a Ellipsoidal CAP

Abstract: 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 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+1-axis 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+1-axis ellip-soid 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/2-arcsinh/a_i, I_1=1-h/a_1 V_(n+1)=Vol(sum from i=1 to n+1(x_i/a_i)~2≤1)is the volume of n+1-axis 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+1-axis ellipsoid in n+1-dimensions Eu-clidean space be then the area formula of ellipsoidal surface is wher…
Key words: ellipsoidal cap; spheral cap; area formula; volume formula; sphere

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