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\centerline{\bf MULTIPLE INTEGRALS}
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\centerline{Hilary Term --- Prof J M Yeomans}

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1. {\bf Repeated integrals}

(a) For the following integrals sketch the region of integration and
so write equivalent integrals with the order of integration reversed.
Evaluate the integrals both ways.
$$
\int_0^{\sqrt{2}} \int_{y^2}^2 y \, dx \, dy  \; , \qquad
\int_0^4  \int_0^{\sqrt{x}} y \sqrt{x}\, dy \, dx \; , \qquad
\int_0^1 \int_{-y}^{y^2}x \, dx \, dy \; .
$$

(b) Reverse the order of integration and hence evaluate:
$$
\int_0^{\pi}  \int_{y}^{\pi} x^{-1} \sin x \, dx \, dy \; .
$$

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2. {\bf Plane polar coordinates and planar mass distributions}

(a) A  mass distribution in the positive $x$ region of the $xy$-plane and 
in the shape of a
semi-circle of radius $a$, centred on the origin, has mass per unit area 
$k$. Find, using plane
polar coordinates,

(i) its mass $M$,
(ii) the coordinates $(\overline{x},\overline{y})$ of its centre of mass,
(iii)  its moments of inertia about the $x$ and $y$ axes.

(b)   Do as above  for a semi-infinite sheet with mass per unit area
$$
\sigma=k \exp-(x^2+y^2)/a^2 \quad \hbox{for} \quad x \geq 0 \; , \qquad
\sigma =0\quad \hbox{for} \quad  x < 0 \; .
$$
where $a$ is a  constant. Comment on the comparisons between the two sets 
of answers.

Note that
$$
\int_0^{\infty} \exp{(-\lambda u^2)}\,du = {1 \over 2} \sqrt{\pi \over 
\lambda} \; .
$$

(c)  Evaluate the following integral:
$$
\int_0^{a} \int_0^{\sqrt{a^2-y^2}}(x^2+y^2)\, \arctan (y/x) \, dx  \, dy \; .
$$

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3. {\bf Jacobian matrix}

The pair of variables $(x,y)$  are each functions of the pair of variables 
$(u,v)$  and
{\it vice versa}.  Consider the matrices
$$
A =\left( \matrix{
     \displaystyle{\partial x\over \partial u} &
\displaystyle{\partial x \over \partial v} \cr
& \cr
     \displaystyle{\partial y\over \partial u} &   \displaystyle{\partial 
y\over \partial v}}
   \right)
\qquad \hbox{and} \qquad
B = \left( \matrix{
    \displaystyle{\partial u\over\partial x} &   \displaystyle{\partial 
u\over\partial y} \cr
& \cr
      \displaystyle{\partial v \over \partial x} &   \displaystyle{\partial 
v \over\partial y}}
  \right) \; .
$$

(a) Show using the chain rule that the product $AB$ of these two matrices 
equals the unit
matrix $I$.

(b) Verify this property explicitly for the case in which $(x,y)$  are 
Cartesian coordinates
and $u$  and $v$  are the polar coordinates $(r,\theta)$.

(c) Assuming the result that the determinant of a matrix  and the 
determinant of its inverse
are reciprocals,  deduce  the relation between the Jacobians
$$   {\partial(u,v)\over \partial(x,y)} =
{\partial u\over\partial x}  {\partial v\over\partial y} -
{\partial u\over\partial y} {\partial v\over\partial x}
\qquad \hbox{and} \qquad
{\partial (x,y)\over\partial (u,v)} =
{\partial x\over\partial u} {\partial y\over\partial v}-
     {\partial x\over\partial v} {\partial y\over\partial u} \; .
$$

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4. {\bf Change of variable in double integral}

(a) Using the change of variable $x+y=u$, $x-y=v$ evaluate the double integral
$ \int\!\!\int_R (x^2 +y^2)\, dx dy $, where $R$ is the region bounded by 
the straight lines $y=x$,
$y=x+2$, $y=-x$ and $y=-x+2$.

(b) Given that $u=xy$ and $v=y/x$,  show that $ 
{\partial(u,v)}/{\partial(x,y)}=
2y/x$. Hence evaluate the integral $$
   \int\!\!\!\int \exp(-xy) \, dxdy
$$
over the region $x>0$, $y>0$, ~$xy<1$, ~$1/2<y/x<2$.

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5. {\bf Jacobian for change from Cartesian to spherical polar \hfill\break
  coordinates}

{\it (This is an alternative derivation of the volume element in spherical 
polars.)}

Spherical polar coordinates are defined in the usual way.   Show that
$$
{\partial(x,y,z)\over \partial(r,\theta,\phi)} = r^2 \sin \theta \; .
$$

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6. {\bf Triple integrals over a hemisphere}

A solid hemisphere of uniform density $k$ occupies the volume
$ x^2 +y^2 +z^2 \leq a^2$, $z\geq 0$.  Using symmetry arguments wherever 
possible, find

(i) its total mass $M$, (ii) the position $(\overline{x}, \overline{y}, 
\overline{z})$ of
its centre-of-mass, and (iii) its moments and products of inertia,
$I_{xx},\;\;I_{yy},\;\;I_{zz},\;\;I_{xy},\;\;I_{yz},\;\;I_{zx},$ where
$$
I_{zz} = \int k \; (x^2 + y^2) \; dV,\;\;\;\;\;\; 
I_{xy} = \int k \;  xy \; dV,\;\;\;\;\;\; \hbox{etc}.
$$

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7. {\bf Surface area of a hemisphere}

Show that the surface area of the curved portion of the hemisphere 
in question (6) is $ 2 \pi
a^2 $ by

(i) directly integrating the element of area $ a^2  \sin \theta
d\theta d\phi $ over the surface of the hemisphere.

(ii) projecting onto an integral taken over the x-y plane.

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8. {\bf Surface integrals}

(a) Find the area of the plane $x-2y+5z=13$ cut out by the cylinder 
$x^2+y^2=9$.

(b) A uniform lamina is made of that part of the plane $x+y+z=1$ which lies 
in the first octant.
Find by integration its area and also its centre of mass.  Use
geometrical arguments to check your result for the area.


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{\bf Answers}


1. (a) 1, 32/5, -1/15; (b) 2.

2.  (a) $\pi k a^2/2$, $(4a/3\pi,0)$, $Ma^2/4$, $Ma^2/4$; (b)
$\pi k a^2/2$, $(a/\sqrt{\pi},0)$, $Ma^2/2$, $Ma^2/2$; (c) $\pi^2 
a^4/32$.

4. (a) 8/3; (b) $(1-e^{-1})\ln 2$.

6. (i) $2\pi a^3 k/3$, (ii) $(0,0,3a/8)$, (iii) $2Ma^2/5$ for all moments 
of inertia and
0 for all products of inertia.

8. (a) $9\pi\sqrt{6/5}$; (b) $\sqrt{3}/2$, $({1/3}, {1/3}, {1/3})$. 

\bye


