4 Engineering Example 3

4.1 Volume of liquid in an ellipsoidal tank

Introduction

An ellipsoidal tank (elliptical when viewed from along $x$ -, $y$ - or $z$ -axes) has a volume of liquid poured into it. It is useful to know in advance how deep the liquid will be. In order to make this calculation, it is necessary to perform a multiple integration and calculate a Jacobian.

Figure 34

Problem in words

The metal tank is in the form of an ellipsoid, with semi-axes $a$ , $b$ and $c$ . A volume $V$ of liquid is poured into the tank ( $V<\frac{4}{3}\pi abc$ , the volume of the ellipsoid) and the problem is to calculate the depth, $h$ , of the liquid.

Mathematical statement of problem

The shaded area is expressed as the triple integral

$V={\int \; <!--nolimits\; -->}_{z=0}^h{\int \; <!--nolimits\; -->}_{y=y_1}^{y_2}{\int \; <!--nolimits\; -->}_{x=x_1}^{x_2}dxdydz$

where limits of integration

$x_1=-a\sqrt{1-\frac{y^2}{b^2}-\frac{{\left(z-c\right)}^2}{c^2}}\text{ and }{x}_2=+a\sqrt{1-\frac{y^2}{b^2}-\frac{{\left(z-c\right)}^2}{c^2}}$

which come from rearranging the equation of the ellipsoid $\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{{\left(z-c\right)}^2}{c^2}=1\right)$ and limits

$y_1=-\frac{b}{c}\sqrt{c^2-{\left(z-c\right)}^2}\text{ and }{y}_2=+\frac{b}{c}\sqrt{c^2-{\left(z-c\right)}^2}$

from the equation of an ellipse in the $y$ - $z$ plane $\left(\frac{y^2}{b^2}+\frac{{\left(z-c\right)}^2}{c^2}=1\right)$ .

Mathematical analysis

To calculate $V$ , use the substitutions

now expressing the triple integral as

$V={\int \; <!--nolimits\; -->}_{z=0}^h{\int \; <!--nolimits\; -->}_{\varphi ={\varphi }_1}^{{\varphi }_2}{\int \; <!--nolimits\; -->}_{\tau ={\tau }_1}^{{\tau }_2}Jd\tau d\varphi dz$

where $J$ is the Jacobian of the transformation calculated from

$J=\left|\begin{array}{ccc}\hfill \frac{\partial x}{\partial \tau }\hfill & \hfill \frac{\partial x}{\partial \varphi }\hfill & \hfill \frac{\partial x}{\partial z}\hfill \\ & & \\ \hfill \frac{\partial y}{\partial \tau }\hfill & \hfill \frac{\partial y}{\partial \varphi }\hfill & \hfill \frac{\partial y}{\partial z}\hfill \\ & & \\ \hfill \frac{\partial z}{\partial \tau }\hfill & \hfill \frac{\partial z}{\partial \varphi }\hfill & \hfill \frac{\partial z}{\partial z}\hfill \\ \hfill \hfill \end{array}\right|$

and reduces to

To determine limits of integration for $\varphi$ , note that the substitutions above are similar to a cylindrical polar co-ordinate system, and so $\varphi$ goes from 0 to $2\pi$ . For $\tau$ , setting $\tau =0$ $\Rightarrow$ $x=0$ and $y=0$ , i.e. the $z$ -axis.

Setting $\tau =1$ gives

$\frac{x^2}{a^2}={cos}^2\varphi \left(1-\frac{{\left(z-c\right)}^2}{c^2}\right)$ (1)

and

$\frac{y^2}{b^2}={sin}^2\varphi \left(1-\frac{{\left(z-c\right)}^2}{c^2}\right)$ (2)

Summing both sides of Equations (1) and (2) gives

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\left({cos}^2\varphi +{sin}^2\varphi \right)\left(1-\frac{{\left(z-c\right)}^2}{c^2}\right)$

or

$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{{\left(z-c\right)}^2}{c^2}=1$

which is the equation of the ellipsoid, i.e. the outer edge of the volume. Therefore the range of $\tau$ should be 0 to 1. Now

Interpretation

Suppose the tank has actual dimensions of $a$ = 2 m, $b$ = 0.5 m and $c$ = 3 m and a volume of $7{\text{m}}^3$ is to be poured into it. (The total volume of the tank is $4\pi m^3\approx 12.57{\text{m}}^3$ ). Then, from above

$V=\frac{\pi ab}{c^2}\left(c{h}^2-\frac{h^3}{3}\right)$

which becomes

$7=\frac{\pi }{9}\left(3h^2-\frac{h^3}{3}\right)$

with solution $h$ = 3.23 m (2 d.p.), compared to the maximum height of the ellipsoid of 6 m.

Exercises

1. The function $f=x^2+y^2$ is to be integrated over an elliptical cone with base being the ellipse, $x^2∕4+y^2=1$ , $z=0$ and apex (point) at $\left(0,0,5\right)$ . The integral can be made simpler by means of the change of variables $x=2\left(1-\frac{w}{5}\right)\tau cos\theta$ , $y=\left(1-\frac{w}{5}\right)\tau sin\theta$ , $z=w$ .

   

   1. Find the limits on the variables $\tau$ , $\theta$ and $w$ .
   2. Find the Jacobian $J\left(\tau ,\theta ,w\right)$ for this transformation.
   3. Express the integral $\int \; <!--nolimits\; -->\int \; <!--nolimits\; -->\int \; <!--nolimits\; -->\left(x^2+y^2\right)dxdydz$ in terms of $\tau$ , $\theta$ and $w$ .
   4. Evaluate this integral. [Hint:- it may be worth noting that ${cos}^2\theta \equiv \frac{1}{2}\left(1+cos2\theta \right)$ ].

      Note: This integral has relevance in topics such as moments of inertia.

2. Using cylindrical polar coordinates, integrate the function $f=z\sqrt{x^2+y^2}$ over the volume between the surfaces $z=0$ and $z=1+x^2+y^2$ for $0\le x^2+y^2\le 1$ .
3. A torus (doughnut) has major radius $R$ and minor radius $r$ . Using the transformation $x=\left(R+\tau cos\alpha \right)cos\theta$ , $y=\left(R+\tau cos\alpha \right)sin\theta$ ,  $z=\tau sin\alpha$ , find the volume of the torus. [Hints:- limits on $\alpha$ and $\theta$ are $0$ to $2\pi$ , limits on $\tau$ are $0$ to $r$ . Show that Jacobian is $\tau \left(R+\tau cos\alpha \right)$ ].

   

4. Find the Jacobian for the following transformations.
   1. $x=u^2+vw$ , $y=2v+u^{2}w$ , $z=uvw$
   2. Cylindrical polar coordinates. $x=\rho cos\theta$ , $y=\rho sin\theta$ , $z=z$

      

Answer