# ANALYTICAL EXPRESSION FOR THE EXACT CURVED SURFACE AREA OF A FRUSTUM OF
HEMIELLIPSOID, THROUGH HYPERGEOMETRIC FUNCTION APPROACH

M.A. Pathan  Centre for Mathematical and Statistical Sciences (CMSS), Peechi,
Thrissur, Kerala-680653, India
Department of Mathematics, Aligarh Muslim University, Aligarh, U.P., India
<EMAIL_ADDRESS>,  Mohd Idris Qureshi  Department of Applied Sciences and
Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (A
Central University), New Delhi 110025, India<EMAIL_ADDRESS>and
Javid Majid∗ Department of Applied Sciences and Humanities, Faculty of
Engineering and Technology, Jamia Millia Islamia (A Central University), New
Delhi 110025, India<EMAIL_ADDRESS>
###### Abstract.

Our present investigation is motivated essentially by several interesting
applications of generalized hypergeometric functions of one, two and more
variables. The hypergeometric functions are potentially useful and have
widespread applications related to the problems in the mathematical, physical,
engineering and statistical sciences. In this article, we aim at obtaining the
analytical expression (not previously found and recorded in the literature)
for the exact curved surface area of a frustum of hemiellipsoid in terms of
Appell’s function of second kind and general triple hypergeometric series of
Srivastava. The derivation is based on Mellin-Barnes type contour integral
representations of generalized hypergeometric function$~{}_{p}F_{q}(z)$,
Meijer’s $G$-function and series manipulation technique. The closed form for
the exact curved surface area of a frustum of hemiellipsoid is also verified
numerically by using Mathematica Program.

###### Key words and phrases:

Meijer’s $G$-function; Mellin-Barnes contour integral; Ellipsoid, Appell’s
function of second kind, General triple hypergeometric function of Srivastava;
Mathematica Program.

###### 2010 Mathematics Subject Classification:

33C20, 33C70, 97G30, 97G40.

∗Corresponding author

##  1\. Introduction and preliminaries

For the definition of Pochhammer symbols, power series form of generalized
hypergeometric function ${}_{p}F_{q}(z)$ and several related results, we refer
the beautiful monographs (see, e.g., [3, 6, 9, 12, 15, 19])

######  Theorem 1.1.

When $p=0,1,2,...$, then

$\frac{1}{\Gamma(\bf{-p})}~{}_{4}F_{3}\left[\begin{array}[]{ll}A,B,C,D;\\\
&z\\\
~{}E,G,\bf{-p};\end{array}\right]=\frac{(A)_{p+1}(B)_{p+1}(C)_{p+1}(D)_{p+1}z^{p+1}}{(E)_{p+1}(G)_{p+1}(p+1)!}\times$
(1.1) $\times~{}_{4}F_{3}\left[\begin{array}[]{ll}A+p+1,B+p+1,C+p+1,D+p+1;\\\
&z\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}E+p+1,~{}G+p+1,~{}2+p;\end{array}\right];~{}|z|<1.$

###### Proof.

Consider the left hand side of the assertion (1.1)

$\frac{1}{\Gamma(-p)}~{}_{4}F_{3}\left[\begin{array}[]{ll}A,B,C,D;\\\ &z\\\
~{}~{}E,G,-p;\end{array}\right]=\sum_{r=0}^{\infty}\frac{(A)_{r}(B)_{r}(C)_{r}(D)_{r}z^{r}}{(E)_{r}(G)_{r}~{}\Gamma(-p+r)~{}r!}$
$=\sum_{r=p+1}^{\infty}\frac{(A)_{r}(B)_{r}(C)_{r}(D)_{r}z^{r}}{(E)_{r}(G)_{r}~{}\Gamma(-p+r)~{}r!}.$

Replacing $r$ by $r+p+1$, after simplification and expressing the result in
the form of generalized hypergeometric function, we arrive at the result
(1.1). ∎

Appell’s function of second kind [19, p.53, Eq.(5)] is defined as:

$F_{2}\left[\begin{array}[]{ll}a;~{}b,c;~{}d,g;~{}x,y\end{array}\right]=F^{1:1;1}_{0:1;1}\left[\begin{array}[]{ll}~{}a:~{}b;~{}c;\\\
&x,y\\\ -:~{}d;~{}g;\\\
\end{array}\right]~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(1.2)
$=\sum_{m,n=0}^{\infty}\frac{(a)_{m+n}(b)_{m}(c)_{n}~{}x^{m}~{}y^{n}}{(d)_{m}(g)_{n}~{}m!~{}n!}=\sum_{n=0}^{\infty}\frac{(a)_{n}(c)_{n}~{}y^{n}}{(g)_{n}~{}n!}~{}{}_{2}F_{1}\left[\begin{array}[]{ll}a+n,~{}b;\\\
&x\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}d;\end{array}\right].$

Convergence conditions of Appell’s double series $F_{2}$:

  1. (i)

Appell’s function $F_{2}$ is convergent when
$|x|+|y|<1;~{}a,b,c,d,g\in\mathbb{C}\backslash\mathbb{Z}^{-}_{0}$.

  2. (ii)

Appell’s function $F_{2}$ is absolutely convergent when $|x|+|y|=1;\\\
~{}x\neq 0,~{}y\neq 0;~{}a,b,c,d,g\in\mathbb{C}\backslash\mathbb{Z}^{-}_{0}$
and $\mathfrak{R}(a+b+c-d-g)<0$.

  3. (iii)

When $a$ is a negative integer, then Appell’s series $F_{2}$ will be a
polynomial, $b,c,d,g\in\mathbb{C}\backslash\mathbb{Z}^{-}_{0}$.

  4. (iv)

When $b,c$ are negative integers, then Appell’s series $F_{2}$ will be a
polynomial, $a,d,g\in\mathbb{C}\backslash\mathbb{Z}^{-}_{0}$.

For absolutely and conditionally convergence of Appell’s double series
$F_{2}$, we refer a beautiful paper of Hài et al.[7].

Mellin-Barnes type contour integral representation of binomial
function$~{}{}_{1}F_{0}(z)$:

(1.3) $(1-z)^{-a}=~{}{}_{1}F_{0}\left[\begin{array}[]{ll}~{}a;\\\ &z\\\
-;\end{array}\right]=\frac{1}{2\pi
i~{}\Gamma(a)}\int_{-i\infty}^{+i\infty}\Gamma(a+s)\Gamma(-s)(-z)^{s}~{}ds:~{}~{}z\neq
0,$

where $|\arg(-z)|<\pi,a\in\mathbb{C}\backslash\mathbb{Z}^{-}_{0}$ and
$i=\sqrt{(-1)}$.

Mellin-Barnes type contour integral representation of $~{}{}_{p}F_{q}(z)$ [19,
p.43, Eq.(6)]:

(1.4)
$~{}{}_{p}F_{q}\left[\begin{array}[]{ll}\alpha_{1},\alpha_{2},...,\alpha_{p};\\\
&z\\\
\beta_{1},\beta_{2},...,\beta_{q};\end{array}\right]=\frac{\Gamma(\beta_{1})\Gamma(\beta_{2})...\Gamma(\beta_{q})}{\Gamma(\alpha_{1})\Gamma(\alpha_{2})...\Gamma(\alpha_{p})}.\frac{1}{2\pi
i}\int_{\sigma-i\infty}^{\sigma+i\infty}\frac{\Gamma(\alpha_{1}+\xi)...\Gamma(\alpha_{p}+\xi)\Gamma(-\xi)(-z)^{\xi}}{\Gamma(\beta_{1}+\xi)...\Gamma(\beta_{q}+\xi)}~{}d\xi,$

where $z\neq 0$.
Convergence conditions:
If $p=q+1$, then $|\arg(-z)|<\pi$.
If $p=q$, then $|\arg(-z)|<\frac{\pi}{2}$,
and no $\alpha_{i}~{}(i=1,2,...,p)$ is zero or negative integer but some of
$\beta_{j}~{}(j=1,2,...,q)$ may be zero or negative integers.

Mellin-Barnes type contour integral representation of Meijer’s G-function
([19, p.45, Eq.(1)], see also [6, 10]):
When $p\leq q$ and $1\leq m\leq q,~{}0\leq n\leq p,$ then

$G^{m,n}_{p,q}\left(z\left|\begin{array}[]{ll}\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{n};\alpha_{n+1},...,\alpha_{p}\\\
\beta_{1},\beta_{2},\beta_{3},...,\beta_{m};\beta_{m+1},...,\beta_{q}\end{array}\right.\right)=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(1.5) $=\frac{1}{2\pi
i}\int_{-i\infty}^{+i\infty}\frac{\Gamma(\beta_{1}-s)...\Gamma(\beta_{m}-s)\Gamma(1-\alpha_{1}+s)...\Gamma(1-\alpha_{n}+s)}{\Gamma(1-\beta_{m+1}+s)...\Gamma(1-\beta_{q}+s)\Gamma(\alpha_{n+1}-s)...\Gamma(\alpha_{p}-s)}(z)^{s}~{}ds,$

where $z\neq 0,(\alpha_{i}-\beta_{j})\neq$ positive integers,
$i=1,2,3,...,n;~{}j=1,2,3,...,m$. For details of contours, see[6, p.207, [10],
p.144].

Relations between Meijer’s $G$\- function and ${}_{2}F_{1}(z)$ [11, p.61,
[20], p.77, Eq.(1)]:

$G^{2~{}2}_{2~{}2}\left(z\left|\begin{array}[]{ll}1-a,1-b;-\\\
0,c-a-b;-\end{array}\right.\right)=\frac{\Gamma(a)\Gamma(b)\Gamma(c-a)\Gamma(c-b)}{\Gamma(c)}{}_{2}F_{1}\left[\begin{array}[]{ll}a,~{}b;\\\
&1-z\\\ ~{}~{}~{}~{}c;\end{array}\right],$

where $|1-z|<1$ and $c-a,c-b\neq 0,-1,-2,...$

$G^{2~{}2}_{2~{}2}\left(z\left|\begin{array}[]{ll}a_{1},a_{2};-\\\
b_{1},b_{2};-\end{array}\right.\right)=\frac{\Gamma(1-a_{1}+b_{1})\Gamma(1-a_{1}+b_{2})\Gamma(1-a_{2}+b_{1})\Gamma(1-a_{2}+b_{2})z^{b_{1}}}{\Gamma(2-a_{1}-a_{2}+b_{1}+b_{2})}\times$
(1.6)
$\times{}_{2}F_{1}\left[\begin{array}[]{ll}1-a_{1}+b_{1},1-a_{2}+b_{1};\\\
&1-z\\\
~{}~{}~{}2-a_{1}-a_{2}+b_{1}+b_{2};\end{array}\right];~{}~{}~{}|1-z|<1.$

A unification of Lauricella’s fourteen triple hypergeometric functions
$F_{1},F_{2},...,F_{14}$[8], including ten functions in Saran’s notations
$F_{E},F_{F},F_{G},F_{K},F_{M},F_{N},F_{P},F_{R},F_{S}$ and $F_{T}$[13, 14]
and the three additional triple hypergeometric functions
$H_{A},H_{B},H_{C}$[16, 17] was introduced by Srivastava [18] who defined a
general triple hypergeometric series $F^{(3)}[x,y,z]$ ([18, p.428], see also
[4, p.156, [5], p.40]):

$F^{(3)}[x,y,z]=F^{(3)}\left[\begin{array}[]{ll}(a_{A})\text{\bf::}~{}(b_{B});(b^{\prime}_{B^{\prime}});~{}(b^{\prime\prime}_{B^{\prime\prime}})\text{\bf:}~{}(c_{C});(c^{\prime}_{C^{\prime}});(c^{\prime\prime}_{C^{\prime\prime}})\text{\bf;}\\\
&x,~{}y,~{}z\\\
(e_{E})\text{\bf::}~{}(g_{G});(g^{\prime}_{G^{\prime}});(g^{\prime\prime}_{G^{\prime\prime}})\text{\bf:}(h_{H});(h^{\prime}_{H^{\prime}});(h^{\prime\prime}_{H^{\prime\prime}})\text{\bf;}\end{array}\right]$
(1.7)
$=\sum_{m,n,p=0}^{\infty}\Lambda(m,n,p)~{}\frac{x^{m}}{m!}~{}\frac{y^{n}}{n!}~{}\frac{z^{p}}{p!},$

where, for convenience,

$\Lambda(m,n,p)=\frac{\prod_{j=1}^{A}(a_{j})_{m+n+p}~{}\prod_{j=1}^{B}(b_{j})_{m+n}~{}\prod_{j=1}^{B^{\prime}}(b^{\prime}_{j})_{n+p}~{}\prod_{j=1}^{B^{\prime\prime}}(b^{\prime\prime}_{j})_{m+p}}{\prod_{j=1}^{E}(e_{j})_{m+n+p}~{}\prod_{j=1}^{G}(g_{j})_{m+n}~{}\prod_{j=1}^{G^{\prime}}(g^{\prime}_{j})_{n+p}~{}\prod_{j=1}^{G^{\prime\prime}}(g^{\prime\prime}_{j})_{m+p}}\times$
(1.8)
$\times\frac{\prod_{j=1}^{C}(c_{j})_{m}~{}\prod_{j=1}^{C^{\prime}}(c^{\prime}_{j})_{n}~{}\prod_{j=1}^{C^{\prime\prime}}(c^{\prime\prime}_{j})_{p}}{\prod_{j=1}^{H}(h_{j})_{m}~{}\prod_{j=1}^{H^{\prime}}(h^{\prime}_{j})_{n}~{}\prod_{j=1}^{H^{\prime\prime}}(h^{\prime\prime}_{j})_{p}}$

where $(a_{A})$ abbreviates the array of $A$ parameters
$a_{1},a_{2},...,a_{A}$, with similar interpretations for
$(b_{B}),(b^{\prime}_{B^{\prime}}),(b^{\prime\prime}_{B^{\prime\prime}}),et~{}cetera$.
The triple hypergeometric series in (1.7) converges when

$\left\\{\begin{array}[]{llrrr}1+E+G+G^{\prime\prime}+H-A-
B-B^{\prime\prime}-C\geq 0,\\\
1+E+G+G^{\prime}+H^{\prime}-A-B-B^{\prime}-C^{\prime}\geq 0,\\\
1+E+G^{\prime}+G^{\prime\prime}+H^{\prime\prime}-A-B^{\prime}-B^{\prime\prime}-C^{\prime\prime}\geq
0,\\\ \end{array}\right.\\\ $

where the equalities hold true for suitably constrained values of $|x|,|y|$
and $|z|$.

Suppose $\phi(x,y)=0$ is the projection of the curved surface of three
dimensional figure $z=f(x,y)$ over the $x$-$y$ plane, then curved surface area
is given by

(1.9) $\hat{S}=\underbrace{\int\int}_{\stackrel{{\scriptstyle\text{over the
area }}}{{\phi(x,y)=0}}}\sqrt{\left\\{1+\left(\frac{\partial z}{\partial
x}\right)^{2}+\left(\frac{\partial z}{\partial
y}\right)^{2}\right\\}}~{}dx~{}dy.$

Figure 1. Projection of curved surface in $x$-$y$ plane.

(1.10) $\text{A definite integral
}\int_{\theta=0}^{\frac{\pi}{2}}\sin^{\alpha}{\theta}\cos^{\beta}{\theta}~{}d\theta=\frac{\Gamma\left(\frac{\alpha+1}{2}\right)\Gamma\left(\frac{\beta+1}{2}\right)}{2\Gamma\left(\frac{\alpha+\beta+2}{2}\right)};~{}\mathfrak{R}(\alpha)>-1,~{}\mathfrak{R}(\beta)>-1.$

Special functions arise in the solution of various classical problems of
physics, generally involving the flow of electromagnetic, acoustic or thermal
energy. In the study of propagation of heat in a metallic bar, one could
consider a bar with a rectangular cross-section, a round cross-section, an
elliptic cross-section or even more complicated cross-sections. In these
situations while dealing with this kind of problem, leads to ordinary
differential equations whose solution forms the majority of special functions
of mathematical physics. In order to determine the allowable frequencies of
oscillation of the waves on the surface of fluid put in a circular dish,
hypergeometric series are required to terminate and determine the allowable
frequencies. The hypergeometric functions have been turned out to have a
variety of applications in a wide range of research subjects (for example see
[1, 2]). In this paper we derive the closed form for obtaining the exact
curved surface area of a frustum of hemiellipsoid by using Mellin-Barnes type
contour integral representations of generalized hypergeometric
function$~{}_{p}F_{q}(z)$ in terms of Appell’s function of second kind and
general triple hypergeometric series of Srivastava in Section 3 with the aid
of some important definite integrals
$\int_{\theta=-\pi}^{\pi}\left(\frac{\cos^{2}{\theta}}{\beta^{2}}+\frac{\sin^{2}{\theta}}{\lambda^{2}}\right)^{s}d\theta$
and $\int_{r=\beta}^{\gamma}\frac{r^{2s+1}}{(1-r^{2})^{s}}dr$ given in Section
2. The problem of approximating the surface area of a triaxial hemiellipsoid
does not appear to have been addressed i.e, no closed form expression exists
for the surface area of a hemiellipsoid. This situation stems from the fact
that it is impossible to execute the integration in the expression for the
surface area in closed form for the most general case of three unequal axes.
Some numerical examples have also been demonstrated in Section 4 among
numerous ones.

##  2\. Evaluation of some useful definite integrals

The following definite integrals hold true associated with suitable
convergence conditions:

######  Theorem 2.1.

(2.1)
$\int_{\theta=-\pi}^{\pi}\left(\frac{\cos^{2}{\theta}}{\sigma^{2}}+\frac{\sin^{2}{\theta}}{\lambda^{2}}\right)^{s}~{}d\theta=\frac{2\pi\lambda}{\sigma^{1+2s}}~{}{}_{2}F_{1}\left[\begin{array}[]{ll}\frac{1}{2},1+s;\\\
&1-\frac{\lambda^{2}}{\sigma^{2}}\\\
~{}~{}~{}~{}~{}~{}~{}~{}1;\end{array}\right],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

where $\sigma\geq\lambda>0$ and it is obvious that
$0\leq(1-\frac{\lambda^{2}}{\sigma^{2}})<1$.

######  Theorem 2.2.

(2.2)
$\int_{\theta=-\pi}^{\pi}\left(\frac{\cos^{2}{\theta}}{\sigma^{2}}+\frac{\sin^{2}{\theta}}{\lambda^{2}}\right)^{s}~{}d\theta=\frac{2\pi\sigma}{\lambda^{1+2s}}~{}{}_{2}F_{1}\left[\begin{array}[]{ll}\frac{1}{2},1+s;\\\
&1-\frac{\sigma^{2}}{\lambda^{2}}\\\
~{}~{}~{}~{}~{}~{}~{}~{}1;\end{array}\right],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

where $\lambda\geq\sigma>0$ and it is obvious that
$0\leq(1-\frac{\sigma^{2}}{\lambda^{2}})<1$.

###### Proof.

For the independent demonstration of the Theorem 2.1

$\text{suppose}~{}I_{1}=\int_{\theta=-\pi}^{\pi}\left(\frac{\cos^{2}{\theta}}{\sigma^{2}}+\frac{\sin^{2}{\theta}}{\lambda^{2}}\right)^{s}~{}d\theta.$

$=\frac{4}{\lambda^{2s}}\int_{\theta=0}^{\frac{\pi}{2}}\left(\sin^{2}\theta\right)^{s}\left\\{1+\frac{\lambda^{2}\cos^{2}{\theta}}{\sigma^{2}\sin^{2}{\theta}}\right\\}^{s}~{}d\theta$
(2.3)
$=\frac{4}{\lambda^{2s}}\int_{\theta=0}^{\frac{\pi}{2}}\sin^{2s}\theta~{}{}_{1}F_{0}\left[\begin{array}[]{ll}~{}~{}-s;\\\
&\frac{-\lambda^{2}\cos^{2}{\theta}}{\sigma^{2}\sin^{2}{\theta}}\\\
~{}-~{}~{};\end{array}\right]~{}d\theta.$

Employing the contour integral (1.3) of ${}_{1}F_{0}(.)$, we get

(2.4) $I_{1}=\frac{2}{\pi
i\Gamma(-s)~{}\lambda^{2s}}\int_{\theta=0}^{\frac{\pi}{2}}\sin^{2s}\theta\left\\{\int_{\zeta=-i\infty}^{+i\infty}\Gamma(-\zeta)\Gamma(-s+\zeta)\left(\frac{\lambda^{2}\cos^{2}{\theta}}{\sigma^{2}\sin^{2}{\theta}}\right)^{\zeta}~{}d\zeta\right\\}~{}d\theta.$

Upon interchanging the order of integration in double integral of (2.4), and
using the formula (1.10), we get

(2.5) $I_{1}=\frac{1}{\pi
i\Gamma(-s)\Gamma(1+s)~{}\lambda^{2s}}\int_{\zeta=-i\infty}^{+i\infty}\Gamma(0-\zeta)\Gamma(1-(s+1)+\zeta)\Gamma\left(\frac{1}{2}+s-\zeta\right)\Gamma\left(1-\frac{1}{2}+\zeta\right)\left(\frac{\lambda^{2}}{\sigma^{2}}\right)^{\zeta}~{}d\zeta.$

Applying Meijer’s $G$-function (1.5), we get

(2.6)
$I_{1}=\frac{2}{\Gamma(-s)\Gamma(1+s)~{}\lambda^{2s}}~{}G^{2~{}2}_{2~{}2}\left(\frac{\lambda^{2}}{\sigma^{2}}\left|\begin{array}[]{ll}s+1,\frac{1}{2};-\\\
0,\frac{1}{2}+s;-\end{array}\right.\right).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Employing the conversion formula (1.6) in equation (2.6), and after further
simplification, we arrive at the result asserted in Theorem 2.1.

The proof of the Theorem 2.2 follows the same steps as in the proof of the
Theorem 2.1. So we omit the details here. ∎

######  Theorem 2.3.

(2.7)
$\int_{r=\beta}^{\gamma}\frac{r^{2s+1}}{(1-r^{2})^{s}}~{}~{}dr=\sum_{\gamma\longrightarrow\beta}^{\circledast}\left\\{\frac{\gamma^{2+2s}}{2(1+s)}~{}{}_{2}F_{1}\left[\begin{array}[]{ll}s,1+s;\\\
&\gamma^{2}\\\
~{}~{}~{}2+s;\end{array}\right]\right\\},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

where $~{}0<\beta<\gamma<1$.

###### Proof.

When $0<\beta<r<\gamma<1$, then

$\int_{r=\beta}^{r=\gamma}\frac{r^{2s+1}}{(1-r^{2})^{s}}~{}~{}dr=\int_{\beta}^{\gamma}r^{2s+1}{(1-r^{2})^{-s}}~{}~{}dr~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}`$

(2.8)
$=\int_{\beta}^{\gamma}r^{2s+1}~{}{}_{1}F_{0}\left[\begin{array}[]{ll}~{}s~{};\\\
&r^{2}\\\ ~{}-;\end{array}\right]~{}dr~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

(2.9)
$=\sum_{p=0}^{\infty}\frac{(s)_{p}~{}}{p!}\int_{\beta}^{\gamma}~{}r^{2p+2s+1}~{}dr~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

(2.10)
$=\frac{1}{2(1+s)}~{}\sum_{p=0}^{\infty}\frac{(s)_{p}~{}(1+s)_{p}~{}}{(2+s)_{p}~{}p!}~{}\left(\gamma^{2p+2s+2}-\beta^{2p+2s+2}\right).$

After further simplification, we arrive at the result as asserted in Theorem
2.3.
∎

Throughout the paper, the semi-axes of the ellipsoid
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$ are assumed in
the form of $a>b>c>0$.

##  3\. Closed form for the curved surface area of a frustum of hemiellipsoid

For the sake of convenience, we shall use the following notation to represent
the lengthy mathematical expression

(3.1)
$\sum_{\gamma\longrightarrow\beta}^{\circledast}\left\\{\Phi(\gamma)\right\\}=\left\\{\Phi(\gamma)\right\\}-\left\\{\Phi(\beta)\right\\}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

######  Theorem 3.1.

The curved surface area of a frustum of hemiellipsoid (whose axis is positive
direction of $z$-axis) i.e,
$z=c\left(1-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\right)^{\frac{1}{2}};a>b>c>0$,
lying above $x$-$y$ plane (i.e, $z=0$) and bounded by the two parallel planes
$z=h$ and $z=H~{}$ is given by:

$\hat{S}~{}=\sum_{\gamma\longrightarrow\beta}^{\circledast}\left\\{b^{2}\gamma^{2}\pi~{}F_{2}\left[1;~{}\frac{1}{2},\frac{-1}{2}~{};~{}1,2~{};1-\frac{b^{2}}{a^{2}},\frac{-c^{2}\gamma^{2}}{a^{2}}\right]+\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(3.2)
$\left.+\frac{b^{2}c^{2}\gamma^{6}\pi}{6a^{2}}~{}F^{(3)}\left[\begin{array}[]{ll}-~{}\text{\bf{:}}\text{\bf{:}}~{}-;2,3;~{}2~{}\text{\bf{:}}~{}\frac{1}{2}~{};~{}1~{};~{}~{}~{}\frac{1}{2}~{}\text{\bf;}\\\
&1-\frac{b^{2}}{a^{2}},\gamma^{2},\frac{-c^{2}\gamma^{2}}{a^{2}}\\\
-~{}\text{\bf{:}}\text{\bf{:}}~{}-;~{}~{}4~{};-~{}\text{\bf{:}}~{}1~{};~{}2~{};~{}2,2~{}\text{\bf;}\\\
\end{array}\right]\right\\},$

where
$0<h<H<c;~{}\beta^{2}={\left(1-\frac{H^{2}}{c^{2}}\right)};~{}\gamma^{2}={\left(1-\frac{h^{2}}{c^{2}}\right)};~{}0<\beta<\gamma<1;~{}|1-\frac{b^{2}}{a^{2}}|<1;~{}a>b>c>0;~{}|\frac{-c^{2}\beta^{2}}{a^{2}}|<1$
and $|\frac{-c^{2}\gamma^{2}}{a^{2}}|<1$.

Remark: The above formula (3.2) is verified numerically through Mathematica
program.

Figure 2. Hemiellipsoid
$z=c\left(1-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\right)^{\frac{1}{2}}$
lying above the $x$-$y$ plane

###### Proof.

The equation of an ellipsoid is given by

(3.3)
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1;~{}~{}a>b>c>0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Intersection of the surface (3.3) with the planes $z=H$ and $z=h$, where
$c>H>h>0$ (parallel to $x$-$y$ plane, lying above $x$-$y$ plane)

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1-\frac{H^{2}}{c^{2}}=\beta^{2}<1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(3.4)
$\text{and}~{}~{}~{}~{}~{}\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1-\frac{h^{2}}{c^{2}}=\gamma^{2}<1,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

where
$\gamma=\sqrt{\left(1-\frac{h^{2}}{c^{2}}\right)},~{}\beta=\sqrt{\left(1-\frac{H^{2}}{c^{2}}\right)}$
and $0<\beta<\gamma<1$.

Again from equation (3.3), we have

(3.5) $\frac{\partial z}{\partial
x}=\frac{-cx}{a^{2}\sqrt{\left(1-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\right)}},~{}\frac{\partial
z}{\partial
y}=\frac{-cy}{b^{2}\sqrt{\left(1-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\right)}}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

The projection of curved surface area of a frustum of hemiellipsoid in $x$-$y$
plane will be the area between the two concentric ellipses, given by.

(3.6)
$\frac{x^{2}}{(a\gamma)^{2}}+\frac{y^{2}}{(b\gamma)^{2}}=1~{}~{}\text{and}~{}~{}\frac{x^{2}}{(a\beta)^{2}}+\frac{y^{2}}{(b\beta)^{2}}=1;~{}~{}a>b,~{}\gamma>\beta$

Figure 3. Two concentric ellipses
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=\gamma^{2},~{}\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=\beta^{2}$.

Substitute the values of $\frac{\partial z}{\partial x}$ and $\frac{\partial
z}{\partial y}$ in equation (1.9). Therefore curved surface area of a frustum
of hemiellipsoid will be

(3.7) $\hat{S}=\underbrace{\int\int}_{\stackrel{{\scriptstyle\text{over the
area between}}}{{\text{ two concentric
ellipses}}}}\sqrt{\left\\{1+\frac{c^{2}x^{2}}{a^{4}\left(1-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\right)}+\frac{c^{2}y^{2}}{b^{4}\left(1-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\right)}\right\\}}~{}dx~{}dy.$

Put $x=aX,~{}y=bY$, therefore,

(3.8) $\hat{S}=ab~{}\underbrace{\int\int}_{\stackrel{{\scriptstyle\text{over
the area between two concentric}}}{{\text{
circles}X^{2}+Y^{2}=\gamma^{2},X^{2}+Y^{2}=\beta^{2}}}}\sqrt{\left\\{1+\frac{c^{2}X^{2}}{a^{2}\left(1-X^{2}-Y^{2}\right)}+\frac{c^{2}Y^{2}}{b^{2}\left(1-X^{2}-Y^{2}\right)}\right\\}}~{}dX~{}dY.$

Figure 4. Concentric circles $r=\gamma,~{}r=\beta$ in polar form

Put $X=r\cos\theta,~{}Y=r\sin\theta$, then $dX~{}dY=rdr~{}d\theta$

(3.9)
$\text{Therefore}~{}\hat{S}=ab~{}\int_{\theta=-\pi}^{\pi}\left(\int_{r=\beta}^{\gamma}\left\\{1+\frac{c^{2}r^{2}}{\left(1-r^{2}\right)}\left(\frac{\cos^{2}\theta}{a^{2}}+\frac{\sin^{2}\theta}{b^{2}}\right)\right\\}^{\frac{1}{2}}~{}rdr\right)~{}d\theta$

Since due to the non availability of a standard formula of definite/indefinite
integrals in the literature of integral calculus for the integration with
respect to $"r"$ and $"\theta"$ in double integral (3.9). We solve such
integrals exactly through hypergeometric function approach.

(3.10)
$\text{Therefore}~{}\hat{S}=ab~{}\int_{\theta=-\pi}^{\pi}\int_{r=\beta}^{\gamma}{}_{1}F_{0}\left[\begin{array}[]{ll}\frac{-1}{2};\\\
&-\frac{c^{2}r^{2}}{\left(1-r^{2}\right)}\left(\frac{\cos^{2}\theta}{a^{2}}+\frac{\sin^{2}\theta}{b^{2}}\right)\\\
~{}-;\end{array}\right]~{}rdr~{}d\theta$

Due to uncertainty in the argument of $~{}_{1}F_{0}(.)$ in equation (3.10), we
apply contour integral (1.3) of ${}_{1}F_{0}(.)$ in equation (3.10), we get

(3.11)
$\hat{S}=ab~{}\int_{\theta=-\pi}^{\pi}\int_{r=\beta}^{\gamma}\left[\frac{1}{2\pi
i\Gamma\left(\frac{-1}{2}\right)}~{}{\int}_{s=-i\infty}^{+i\infty}~{}\Gamma(-s)\Gamma(\frac{-1}{2}+s)\left\\{\frac{c^{2}r^{2}}{(1-r^{2})}\left(\frac{\cos^{2}\theta}{a^{2}}+\frac{\sin^{2}\theta}{b^{2}}\right)\right\\}^{s}ds\right]~{}rdr~{}d\theta,$

where
$|\arg\left\\{c^{2}\left(\frac{\cos^{2}\theta}{a^{2}}+\frac{\sin^{2}\theta}{b^{2}}\right)\right\\}|<\pi$.

Interchanging the order of integration in double integral of (3.11)and
employing the formulas (2.1) and (2.7), we get

$\hat{S}=\sum_{\gamma\longrightarrow\beta}^{\circledast}\left\\{\frac{-b^{2}\gamma^{2}}{4\sqrt{\pi}~{}i}~{}{\int}_{s=-i\infty}^{+i\infty}~{}\frac{\Gamma(-s)\Gamma(\frac{-1}{2}+s)}{(1+s)}\left(\frac{c\gamma}{a}\right)^{2s}~{}{}_{2}F_{1}\left[\begin{array}[]{ll}\frac{1}{2},1+s;\\\
&1-\frac{b^{2}}{a^{2}}\\\
~{}~{}~{}~{}~{}~{}~{}~{}1;\end{array}\right]\times\right.$ (3.12)
$\left.\times~{}{}_{2}F_{1}\left[\begin{array}[]{ll}s,1+s;\\\ &\gamma^{2}\\\
~{}~{}~{}2+s;\end{array}\right]ds\right\\},$

where $|\gamma^{2}|<1$ and $~{}|1-\frac{b^{2}}{a^{2}}|<1$.

$\hat{S}=\sum_{\gamma\longrightarrow\beta}^{\circledast}\left\\{\frac{-b^{2}\gamma^{2}}{4\sqrt{\pi}~{}i}~{}{\int}_{s=-i\infty}^{+i\infty}~{}\frac{\Gamma(-s)\Gamma(\frac{-1}{2}+s)}{(1+s)}\left(\frac{c\gamma}{a}\right)^{2s}~{}\left(\sum_{m=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{m}(1+s)_{m}\left(1-\frac{b^{2}}{a^{2}}\right)^{m}}{(1)_{m}~{}m!}\right)~{}ds-\frac{b^{2}\gamma^{2}}{4\sqrt{\pi}~{}i}\times\right.$
(3.13)
$\left.\times{\int}_{s=-i\infty}^{+i\infty}~{}\frac{\Gamma(-s)\Gamma(\frac{-1}{2}+s)}{(1+s)}\left(\frac{c\gamma}{a}\right)^{2s}~{}\left(\sum_{m=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{m}(1+s)_{m}\left(1-\frac{b^{2}}{a^{2}}\right)^{m}}{(1)_{m}~{}m!}\right)\left(\sum_{n=1}^{\infty}\frac{(s)_{n}(1+s)_{n}~{}\gamma^{2n}}{(2+s)_{n}~{}n!}\right)~{}ds\
\right\\}.$

On interchanging the order of summation and integration, we get

$\hat{S}=\sum_{\gamma\longrightarrow\beta}^{\circledast}\left\\{\frac{-b^{2}\gamma^{2}}{4\sqrt{\pi}~{}i}~{}\sum_{m=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{m}\left(1-\frac{b^{2}}{a^{2}}\right)^{m}}{(1)_{m}~{}m!}{\int}_{s=-i\infty}^{+i\infty}~{}\frac{\Gamma(\frac{-1}{2}+s)\Gamma(1+m+s)\Gamma(-s)}{\Gamma(2+s)}\left(\frac{c^{2}\gamma^{2}}{a^{2}}\right)^{s}~{}ds-\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$-\frac{b^{2}\gamma^{2}}{4\sqrt{\pi}~{}i}\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}\frac{\left(\frac{1}{2}\right)_{m}\left(1-\frac{b^{2}}{a^{2}}\right)^{m}\gamma^{2n}}{(1)_{m}~{}m!~{}n!}\times~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(3.14)
$\left.\times{\int}_{s=-i\infty}^{+i\infty}~{}\frac{\Gamma(\frac{-1}{2}+s)\Gamma(1+m+s)\Gamma(n+s)\Gamma(1+n+s)\Gamma(-s)}{\Gamma(0+s)\Gamma(1+s)\Gamma(2+n+s)}\left(\frac{c^{2}\gamma^{2}}{a^{2}}\right)^{s}~{}ds\right\\}.$

Replacing $n$ by $n+1$ in equation (3.14) and applying contour integral (1.4),
we get

$\hat{S}=\sum_{\gamma\longrightarrow\beta}^{\circledast}\left\\{b^{2}\gamma^{2}\pi\sum_{m=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{m}\left(1-\frac{b^{2}}{a^{2}}\right)^{m}}{(1)_{m}}~{}_{2}{F}_{1}\left[\begin{array}[]{lll}\frac{-1}{2},1+m;\\\
&\frac{-c^{2}\gamma^{2}}{a^{2}}\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2~{};\\\
\end{array}\right]+\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(3.15)
$\left.+b^{2}\gamma^{4}\pi\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{m}\left(1-\frac{b^{2}}{a^{2}}\right)^{m}~{}\gamma^{2n}~{}\Gamma(1+n)}{(1)_{m}~{}\Gamma(3+n)~{}}~{}\left(\frac{1}{\Gamma(0)}~{}_{4}{F}_{3}\left[\begin{array}[]{lll}\frac{-1}{2},1+m,1+n,2+n;\\\
&\frac{-c^{2}\gamma^{2}}{a^{2}}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0,~{}1,~{}3+n~{};\\\
\end{array}\right]\right)\right\\},$

where $\frac{-c^{2}\gamma^{2}}{a^{2}}<1.$

On further simplification, we arrive at the result as asserted in Theorem 3.1
with the aid of the assertion given in (1.1). ∎

##  4\. Applications of the closed form (3.2)

To understand the subject matter, we shall discuss some numerical problems:

Problem 1. Find the exact curved surface area of a frustum of hemiellipsoid
$z=\left(1-\frac{x^{2}}{9}-\frac{y^{2}}{4}\right)^{\frac{1}{2}}$ bounded by
the planes $z=\sqrt{0.6}$ and $z=\sqrt{0.2}$.

Solution:  We have $a=3,~{}b=2,~{}c=1,~{}H=\sqrt{0.6}=0.774596669...\text{
and}~{}h=\sqrt{0.2}=0.447213595...,$ then $\beta=0.2\text{ and }\gamma=0.6$

Substituting the above values of $a,b,c,h,H,\beta$ and $\gamma$ in the closed
form (3.2), we have

$\hat{S}=(1.44\pi)F_{2}\left[1;\frac{1}{2},\frac{-1}{2};1,2;0.55...,-0.04\right]-(0.16\pi)F_{2}\left[1;\frac{1}{2},\frac{-1}{2};1,2;0.55...,-0.0044...\right]+$
$+(0.003456\pi)~{}F^{(3)}\left[\begin{array}[]{ll}-~{}\text{\bf{:}}\text{\bf{:}}~{}-;2,3;~{}2~{}\text{\bf{:}}~{}\frac{1}{2}~{};~{}1~{};~{}~{}~{}\frac{1}{2}~{}\text{\bf;}\\\
&0.55...\text{\bf,}~{}0.36\text{\bf,}~{}0.04\\\
-~{}\text{\bf{:}}\text{\bf{:}}~{}-;~{}~{}4~{};-~{}\text{\bf{:}}~{}1~{};~{}2~{};~{}2,2~{}\text{\bf;}\\\
\end{array}\right]-$ (4.1)
$-(0.000004740...\pi)~{}F^{(3)}\left[\begin{array}[]{ll}-~{}\text{\bf{:}}\text{\bf{:}}~{}-;2,3;~{}2~{}\text{\bf{:}}~{}\frac{1}{2}~{};~{}1~{};~{}~{}~{}\frac{1}{2}~{}\text{\bf;}\\\
&0.55...\text{\bf,}~{}0.04\text{\bf,}~{}0.0044...\\\
-~{}\text{\bf{:}}\text{\bf{:}}~{}-;~{}~{}4~{};-~{}\text{\bf{:}}~{}1~{};~{}2~{};~{}2,2~{}\text{\bf;}\\\
\end{array}\right].$

Now using the Mathematica program to calculate the sum of the double and
triple infinite series in equation (4.1), we get the exact curved surface area
of a frustum of hemiellipsoid
$z=\left(1-\frac{x^{2}}{9}-\frac{y^{2}}{4}\right)^{\frac{1}{2}}$ bounded by
the planes $z=\sqrt{0.6}$ and $z=\sqrt{0.2}$ is
$\hat{S}=6.1749238...~{}\text{square units.}$

Problem 2. Find the exact curved surface area of a frustum of hemiellipsoid
$z=2\left(1-\frac{x^{2}}{25}-\frac{y^{2}}{9}\right)^{\frac{1}{2}}$ bounded by
the planes $z=\sqrt{3.64}$ and $z=\sqrt{1.44}$.

Solution:  We have $a=5,~{}b=3,~{}c=2,~{}H=\sqrt{3.64}=1.9078784...\text{
and}~{}h=\sqrt{1.44}=1.2,$ then $\beta=0.3\text{ and }\gamma=0.8$

Substituting the above values of $a,b,c,h,H,\beta$ and $\gamma$ in the closed
form (3.2), we have

$\hat{S}=(5.76\pi)F_{2}\left[1;\frac{1}{2},\frac{-1}{2};1,2;0.64,-0.1024\right]-(0.81\pi)F_{2}\left[1;\frac{1}{2},\frac{-1}{2};1,2;0.64,-0.0144\right]+$
$+(0.0629146\pi)~{}F^{(3)}\left[\begin{array}[]{ll}-~{}\text{\bf{:}}\text{\bf{:}}~{}-;2,3;~{}2~{}\text{\bf{:}}~{}\frac{1}{2}~{};~{}1~{};~{}~{}~{}\frac{1}{2}~{}\text{\bf;}\\\
&0.64\text{\bf,}~{}0.64\text{\bf,}~{}0.1024\\\
-~{}\text{\bf{:}}\text{\bf{:}}~{}-;~{}~{}4~{};-~{}\text{\bf{:}}~{}1~{};~{}2~{};~{}2,2~{}\text{\bf;}\\\
\end{array}\right]-$ (4.2)
$-(0.00017496\pi)~{}F^{(3)}\left[\begin{array}[]{ll}-~{}\text{\bf{:}}\text{\bf{:}}~{}-;2,3;~{}2~{}\text{\bf{:}}~{}\frac{1}{2}~{};~{}1~{};~{}~{}~{}\frac{1}{2}~{}\text{\bf;}\\\
&0.64\text{\bf,}~{}0.09\text{\bf,}~{}0.0144\\\
-~{}\text{\bf{:}}\text{\bf{:}}~{}-;~{}~{}4~{};-~{}\text{\bf{:}}~{}1~{};~{}2~{};~{}2,2~{}\text{\bf;}\\\
\end{array}\right].$

Now using the Mathematica program to calculate the sum of the double and
triple infinite series in equation (4.2), we get the exact curved surface area
of a frustum of hemiellipsoid
$z=2\left(1-\frac{x^{2}}{25}-\frac{y^{2}}{9}\right)^{\frac{1}{2}}$ bounded by
the planes $z=\sqrt{3.64}$ and $z=\sqrt{1.44}$ is
$\hat{S}=28.41904...~{}\text{square units.}$

##  5\. Conclusion

In this paper, we obtained the closed form for the exact curved surface area
of a frustum of hemiellipsoid
$z=c\left(1-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\right)^{\frac{1}{2}}$
obtained by the interception of a hemiellipsoid with the two parallel planes
$z=h$ and $z=H~{}(H>h)$, through hypergeometric function approach i.e, by
using series rearrangement technique, Mellin-Barnes type contour integral
representations of generalized hypergeometric function$~{}_{p}F_{q}(z)$ and
Meijer’s $G$-function; in terms of Appell’s function of second kind and
general triple hypergeometric series of Srivastava. The formula is neither
available in the literature of mathematics nor found in any mathematical
tables. We conclude that several formulas for exact curved surface area of
frustum of other three dimensional figures can be derived in an analogous
manner, using Mellin-Barnes contour integration. Moreover, the results deduced
above (presumably new), have potential applications in the fields of applied
mathematics, statistics and engineering sciences.

## Disclosure statement

No potential conflict of interest was reported by the authors.

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