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\centerline{{\byg MA21}}
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\centerline{{\bold Sheet Ten}}
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{\menu (To be handed in to the box outside McCrea 232 on
the morning of the first lecturing day of next Term i.e.{\bold 
Wednesday 13th January}).}
\hfill\break
\vfill
\noindent
(1) \quad Evaluate
$$\int\limits_{y=0}^1dy\,\int\limits_{x=3y}^3 e^{x^2}dx.$$
(Hint: first find the region over which the integral is taken, and then 
reverse the order of integration.)
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\noindent
(2) \quad Show that
$$\int\!\!\!\int_Rxy^2 dx\,dy$$
where $R$ is the smaller segment of the circle $x^2+y^2=a^2$ cut off by the
line $x+y=a$ is equal to $\displaystyle{a^5\over 20}$.
\bigskip
\noindent
(3) \quad Evaluate
$$\int\!\!\!\int_Rxy dx\,dy$$
where $R$ is the quadrant of the circle $x^2+y^2=a^2$ for which $x>0$, $y>0$.
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(Hint: use polar coordinates.)
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\noindent
(4) \quad Evaluate
$$\int\!\!\!\int_Re^{-\left(x^2+y^2\right)}dx\,dy$$
where $R$ is the infinite quadrant given by $x>0$, $y>0$, by substituting
plane polar coordinates.
\par
Let $I=\int\limits _0^\infty e^{-x^2}dx$. Evaluate $I^2$. (Hint: remember
that $\int\limits _0^\infty e^{-x^2}dx$ is equal to 
$\int\limits _0^\infty e^{-y^2}dy$, and use the result of the first part.)
\par
Hence show that
$$\int_0^\infty e^{-x^2}dx={1\over 2}\sqrt\pi.$$
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\rightline{Peter Rado}
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