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diff --git a/dviware/crudetype/version3/rado.tex b/dviware/crudetype/version3/rado.tex new file mode 100644 index 0000000000..cfccd49e46 --- /dev/null +++ b/dviware/crudetype/version3/rado.tex @@ -0,0 +1,64 @@ +%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 |