An ellipsoidal tank (elliptical when viewed from along -, - or -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.
Problem in words
The metal tank is in the form of an ellipsoid, with semi-axes , and . A volume of liquid is poured into the tank ( , the volume of the ellipsoid) and the problem is to calculate the depth, , of the liquid.
Mathematical statement of problem
The shaded area is expressed as the triple integral
where limits of integration
which come from rearranging the equation of the ellipsoid and limits
from the equation of an ellipse in the - plane .
To calculate , use the substitutions
now expressing the triple integral as
where is the Jacobian of the transformation calculated from
and reduces to
To determine limits of integration for , note that the substitutions above are similar to a cylindrical polar co-ordinate system, and so goes from 0 to . For , setting and , i.e. the -axis.
Summing both sides of Equations (1) and (2) gives
which is the equation of the ellipsoid, i.e. the outer edge of the volume. Therefore the range of should be 0 to 1. Now
Suppose the tank has actual dimensions of = 2 m, = 0.5 m and = 3 m and a volume of is to be poured into it. (The total volume of the tank is ). Then, from above
with solution = 3.23 m (2 d.p.), compared to the maximum height of the ellipsoid of 6 m.
is to be integrated over an elliptical cone with base being the ellipse,
and apex (point) at
. The integral can be made simpler by means of the change of variables
- Find the limits on the variables , and .
- Find the Jacobian for this transformation.
- Express the integral in terms of , and .
Evaluate this integral. [Hint:- it may be worth noting that
Note: This integral has relevance in topics such as moments of inertia.
- Using cylindrical polar coordinates, integrate the function over the volume between the surfaces and for .
A torus (doughnut) has major radius
and minor radius
. Using the transformation
, find the volume of the torus. [Hints:- limits on
, limits on
. Show that Jacobian is
Find the Jacobian for the following transformations.
- , ,
Cylindrical polar coordinates.
- : to , : to , : to