Consider the 2D region 𝐴 which represents part of the cross-section of the lens in the eye’s pupil (Figure
3). The area of 𝐴 is given by the sum of the areas of three subregions 𝐴”, 𝐴!, 𝐴# as follows:
𝐴 = 𝐴”+𝐴!+𝐴#
𝐴” = 2 2 𝑥𝑦𝑑𝑦𝑑𝑥
%&’
%&√”)’!
‘&”
‘& ”
√!
; 𝐴! = 2 2 𝑥𝑦𝑑𝑦𝑑𝑥
%&’
%&*
‘&√!
‘&”
; 𝐴# = 2 2 𝑥𝑦𝑑𝑦𝑑𝑥
%&√+)’!
%&*
‘&!
‘&√!
a) Sketch 𝐴, showing its three different subregions 𝐴”, 𝐴!, 𝐴#.
b) Express 𝐴 as one double integral. Evaluate this integral to find 𝐴. Check
A using MATLAB.
c) Consider the sclera in Figure 3: the dense connective tissue of the eyeball that forms the “white”
of the eye. Compute an approximation 𝑀 for the mass of the sclera, assuming the sclera is a thin
spherical shell centred at the origin with inner radius 𝑟 = 𝑎 and outer radius 𝑟 = 𝑏. Assume
further that the sclera density 𝜌 is given by
𝜌(𝑥, 𝑦, 𝑧) = ”
(‘!-%!-.!)”/!
where 𝑛 is a positive integer such that 𝑛 ≠ 3 and 0 < 𝑎 < 𝑏.

d) Glaucoma, the leading cause of irreversible blindness worldwide, is an eye disease where the so-
called humour fluid expands and covers the optic nerve, which in turn leads to vision loss.
The divergence and curl of the humour fluid flow velocity 𝑽 are key components of models of
humour fluid dynamics (Figure 4).
i. Show that 𝑽(𝑥, 𝑦, 𝑧), given by
𝑉 = ( 2𝑥𝑦𝑧# + 𝑦𝑒’% ) 𝐢 + ( 𝑥𝑒’% + 𝑥!𝑧#) 𝐣 + ( 3𝑥!𝑦 𝑧! + cos 𝑧 ) 𝐤 ,
is conservative.
ii. Find a scalar function 𝜙(𝑥, 𝑦, 𝑧), such that 𝑽 = ∇𝜙.
iii. Determine the work done by 𝑽 to move a particle fluid along the finite curve 𝑪 with
parametrization
𝐫(𝑡) = (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) = (𝑡, 𝑡 sin 𝑡 , (𝑡 + 1) cos 𝑡), 0 ≤ 𝑡 ≤ 2𝜋.
Sketch 𝑪 using MATLAB.
iv. Find the divergence of a fluid velocity 𝑽, given
𝑽 = 𝑥𝑦!𝑧 𝐢 − 𝑥𝑧! 𝐣 + 𝑥!𝑦 𝐤.
State the values of x, y and z where the point (𝑥, 𝑦, 𝑧) is a sink of 𝑽.