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+%This is for the maths printer
+\magnification = \magstephalf
+\vsize=10 truein \hsize=170truemm
+\voffset=0truein \hoffset=0truecm
+
+\parskip = 1ex plus .5ex minus .1ex
+\parindent 3em
+\nopagenumbers
+
+\font\Byg = cmr10 at 17.28 truept
+\font\byg = cmr10 at 14.40 truept
+\font\menu = cmti10 at 10.00 truept
+\font\small = cmr10 at 10.00 truept
+\font\bold = cmb10
+\font\bcaps = cmssbx10
+\font\pfont = cmu10
+\font\tiny = cmr6 at 6 truept
+\def\pound{{\pfont \$}}
+
+\centerline{{\byg MA21}}
+\vskip .75 truein
+\centerline{{\bold Sheet Ten}}
+\vskip 0.2 truein
+{\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.)
+\hfill
+\bigskip
+\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 $x>0$, $y>0$.
+\hfill\break
+(Hint: use polar coordinates.)
+\hfill
+\bigskip
+\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
+$$\int_0^\infty e^{-x^2}dx={1\over 2}\sqrt\pi.$$
+\vfill
+\rightline{Peter Rado}
+\end