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{\bf Exam Question
Topic: SurfaceIntegral} The dean's trophy presented to the best
lecturer of the year is in the shape of the upper half of a
cylinder, specified by $$y^2+z^2=1,\ z\ge0,\ 0\le x\le1.$$ The
curved part of the surface is to be covered with an iridescent
foil made of a mixture of precious metals, where the composition
varies so as to achieve a colour change over the surface. The
density at a point $(x,y,z)$ on the surface is given by
$(1+2x^2).$
Calculate the total mass of foil by evaluating an appropriate
surface integral. \vspace{0.5in}
{\bf Solution}
The equation of the surface can be rewritten as $z=\sqrt{1-y^2},$
so $$\frac{\partial z}{\partial y}=\frac{-y}{\sqrt{1-y^2}};\
\frac{\partial z}{\partial x}=0. $$
Therefore we have $$dS = \sqrt{1+\left(\frac{\partial z}{\partial
x}\right)^2+\left(\frac{\partial z}{\partial
y}\right)^2}=\sqrt{1+\frac{y^2}{1-y^2}}=\frac{1}{\sqrt{1-y^2}}$$
The mass is therefore given by
\begin{eqnarray*}
M &=&\int_0^1\, dx\int_{-1}^1 \frac{1+2x^2}{\sqrt{1-y^2}}\, dy
=\int_0^1(1+2x^2)\, dx\int_{-1}^1\frac{1}{\sqrt{1-y^2}}\, dy\\ &=&
\left[x+\frac{2}{3}x^3\right]_0^1
\left[\sin^{-1}y\right]_{-1}^1=\frac{5\pi}{3}.
\end{eqnarray*}
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