(*^ ::[paletteColors = 128; fontset = title, "Times", 24, L2, center, bold, nohscroll; fontset = subtitle, "Times", 18, L2, center, bold, nohscroll; fontset = subsubtitle, "Times", 14, L2, center, italic, nohscroll; fontset = section, "Times", 18, L2, bold, nohscroll, grayBox; fontset = subsection, "Times", 14, L2, bold, nohscroll, blackBox; fontset = subsubsection, "Times", 12, L2, bold, nohscroll, whiteBox; fontset = text, "Times", 12, L2, nohscroll; fontset = smalltext, "Times", 10, L2, nohscroll; fontset = input, "Courier", 12, L2, bold, nowordwrap; fontset = output, "Courier", 12, L2, nowordwrap; fontset = message, "Courier", 12, L2, R65535, nowordwrap; fontset = print, "Courier", 12, L2, nowordwrap; fontset = info, "Courier", 12, L2, B65535, nowordwrap; fontset = postscript, "Courier", 12, L2, nowordwrap; fontset = name, "Geneva", 10, L2, italic, nohscroll; fontset = header, "Times", 12, L2; fontset = footer, "Times", 12, L2, center; fontset = help, "Times", 10, L2, nohscroll; 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.22421 .08485 L .22612 .09366 L .22803 .10134 L .22994 .11091 L .23184 .1075 L .23375 .10159 L .23566 .10098 L .23757 .10605 L .23948 .11705 L .24139 .1149 L .2433 .11485 L .2452 .12209 L .24711 .12609 L .24902 .12669 L .25093 .12147 L .25284 .11663 L .25475 .11061 L .25666 .11198 L .25856 .10697 L .26047 .11001 L .26238 .11773 L .26429 .11954 L .2662 .12379 L .26811 .11634 L .27002 .11796 L .27192 .12037 L .27383 .10701 L .27574 .09874 L .27765 .09123 L .27956 .09529 L .28147 .09117 L .28338 .08877 L .28528 .09238 L .28719 .09233 L .2891 .08468 L .29101 .08783 L .29292 .08762 L .29483 .08665 L .29674 .07689 L .29864 .07627 L .30055 .08339 L .30246 .08544 L .30437 .08017 L .30628 .08675 L .30819 .09269 L Mistroke .3101 .09429 L .312 .09308 L .31391 .0962 L .31582 .09784 L .31773 .09349 L .31964 .09656 L .32155 .09504 L .32346 .09504 L .32537 .09472 L .32727 .09211 L .32918 .08769 L .33109 .08838 L .333 .09108 L .33491 .08784 L .33682 .09514 L .33873 .09323 L .34063 .09294 L .34254 .0956 L .34445 .09492 L .34636 .09059 L .34827 .07998 L .35018 .0807 L .35209 .08333 L .35399 .08347 L .3559 .0866 L .35781 .09058 L .35972 .09352 L .36163 .09124 L .36354 .08475 L .36545 .07885 L .36735 .0788 L .36926 .0757 L .37117 .08162 L .37308 .07269 L .37499 .06755 L .3769 .06976 L .37881 .07446 L .38071 .06691 L .38262 .0675 L .38453 .06394 L .38644 .0576 L .38835 .05658 L .39026 .0596 L .39217 .05808 L .39407 .05903 L .39598 .06272 L .39789 .06555 L .3998 .05717 L .40171 .05182 L .40362 .04805 L Mistroke .40553 .05285 L .40743 .05357 L .40934 .05196 L .41125 .05124 L .41316 .04799 L .41507 .05164 L .41698 .05473 L .41889 .05546 L .42079 .05745 L .4227 .05936 L .42461 .05694 L .42652 .05595 L .42843 .05883 L .43034 .05388 L .43225 .05225 L .43415 .05798 L .43606 .05008 L .43797 .04474 L .43988 .04985 L .44179 .05185 L .4437 .05451 L .44561 .05594 L .44751 .05623 L .44942 .05787 L .45133 .05994 L .45324 .06198 L .45515 .06048 L .45706 .0606 L .45897 .06403 L .46087 .06033 L .46278 .05659 L .46469 .05701 L .4666 .06236 L .46851 .06565 L .47042 .06544 L .47233 .06723 L .47423 .06634 L .47614 .06259 L .47805 .0627 L .47996 .06047 L .48187 .05794 L .48378 .05655 L .48569 .05936 L .48759 .05552 L .4895 .05707 L .49141 .05156 L .49332 .05448 L .49523 .05647 L .49714 .05691 L .49905 .05535 L Mistroke .50095 .05598 L .50286 .05319 L .50477 .0558 L .50668 .05459 L .50859 .05309 L .5105 .05436 L .51241 .05711 L .51431 .05672 L .51622 .0564 L .51813 .0565 L .52004 .05854 L .52195 .0587 L .52386 .06118 L .52577 .06006 L .52767 .06347 L .52958 .05982 L .53149 .06299 L .5334 .06397 L .53531 .06603 L .53722 .06758 L .53913 .06575 L .54103 .06142 L .54294 .06173 L .54485 .06083 L .54676 .06232 L .54867 .06276 L .55058 .06276 L .55249 .06167 L .55439 .05573 L .5563 .05855 L .55821 .05544 L .56012 .05196 L .56203 .05717 L .56394 .06044 L .56585 .06563 L .56775 .06726 L .56966 .06569 L .57157 .06581 L .57348 .06901 L .57539 .07015 L .5773 .06908 L .57921 .06706 L .58111 .06727 L .58302 .06782 L .58493 .0698 L .58684 .07199 L .58875 .069 L .59066 .07274 L .59257 .07415 L .59447 .06964 L Mistroke .59638 .067 L .59829 .06879 L .6002 .06826 L .60211 .06259 L .60402 .05961 L .60593 .05851 L .60783 .05958 L .60974 .05584 L .61165 .05887 L .61356 .06 L .61547 .05806 L .61738 .06028 L .61929 .06237 L .62119 .06497 L .6231 .06418 L .62501 .0612 L .62692 .06249 L .62883 .05887 L .63074 .06263 L .63265 .05874 L .63455 .06162 L .63646 .05961 L .63837 .06168 L .64028 .06327 L .64219 .06378 L .6441 .06565 L .64601 .06597 L .64791 .06305 L .64982 .05975 L .65173 .05875 L .65364 .05901 L .65555 .06075 L .65746 .06002 L .65937 .06113 L .66127 .06141 L .66318 .06007 L .66509 .05508 L .667 .05642 L .66891 .0563 L .67082 .05944 L .67273 .06028 L .67463 .05925 L .67654 .05711 L .67845 .05858 L .68036 .06087 L .68227 .05799 L .68418 .06045 L .68609 .05887 L .688 .06009 L .6899 .06102 L Mistroke .69181 .06149 L .69372 .06341 L .69563 .06054 L .69754 .05903 L .69945 .0583 L .70136 .05976 L .70326 .06076 L .70517 .06185 L .70708 .06271 L .70899 .06356 L .7109 .06233 L .71281 .05819 L .71472 .05772 L .71662 .05944 L .71853 .05902 L .72044 .05962 L .72235 .06098 L .72426 .06134 L .72617 .05724 L .72808 .05537 L .72998 .05838 L .73189 .05583 L .7338 .05661 L .73571 .05743 L .73762 .05568 L .73953 .05552 L .74144 .05755 L .74334 .05901 L .74525 .05831 L .74716 .05751 L .74907 .05492 L .75098 .055 L .75289 .05141 L .7548 .05038 L .7567 .04963 L .75861 .04675 L .76052 .0465 L .76243 .04688 L .76434 .04836 L .76625 .04768 L .76816 .04768 L .77006 .04583 L .77197 .04651 L .77388 .04577 L .77579 .04527 L .7777 .04378 L .77961 .04266 L .78152 .04231 L .78342 .04021 L .78533 .04217 L Mistroke .78724 .04141 L .78915 .03977 L .79106 .04156 L .79297 .04204 L .79488 .04369 L .79678 .04125 L .79869 .04079 L .8006 .03673 L .80251 .03768 L .80442 .0385 L .80633 .04014 L .80824 .04076 L .81014 .04276 L .81205 .0468 L .81396 .04721 L .81587 .04989 L .81778 .05131 L .81969 .05146 L .8216 .0514 L .8235 .04797 L .82541 .04886 L .82732 .05109 L .82923 .05089 L .83114 .0478 L .83305 .04753 L .83496 .05036 L .83686 .05241 L .83877 .05254 L .84068 .05174 L .84259 .04929 L .8445 .05021 L .84641 .04918 L .84832 .0481 L .85022 .04605 L .85213 .04659 L .85404 .04688 L .85595 .04825 L .85786 .0504 L .85977 .04969 L .86168 .04942 L .86358 .04947 L .86549 .05115 L .8674 .05365 L .86931 .05201 L .87122 .04922 L .87313 .04928 L .87504 .0502 L .87694 .05123 L .87885 .05157 L .88076 .05081 L Mistroke .88267 .05041 L .88458 .05217 L .88649 .05208 L .8884 .05278 L .8903 .05303 L .89221 .054 L .89412 .05286 L .89603 .05409 L .89794 .05345 L .89985 .0525 L .90176 .05257 L .90366 .05259 L .90557 .04945 L .90748 .05066 L .90939 .0533 L .9113 .05492 L .91321 .05415 L .91512 .05585 L .91702 .05685 L .91893 .05527 L .92084 .05686 L .92275 .05878 L .92466 .06168 L .92657 .05824 L .92848 .05805 L .93038 .05892 L .93229 .05624 L .9342 .05796 L .93611 .0602 L .93802 .061 L .93993 .06119 L .94184 .06224 L .94374 .06296 L .94565 .06282 L .94756 .06199 L .94947 .0615 L .95138 .0633 L .95329 .06431 L .9552 .06561 L .9571 .06355 L .95901 .065 L .96092 .06556 L .96283 .06628 L .96474 .06281 L .96665 .0606 L .96856 .06207 L .97046 .06312 L .97237 .06153 L .97428 .06064 L .97619 .06115 L Mistroke Mfstroke P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = input; endGroup; preserveAspect; ] :[font = subsection; inactive; startGroup; preserveAspect; ] Figure 5 :[font = input; preserveAspect; ] F[x_] := Pi x^2/4 /; (x >=0 && x <= 1); F[x_] := Sqrt[x^2 - 1] + Pi x^2/4 - x^2 ArcCos[1/x] /; (x > 1 && x <= N[Sqrt[2]]) :[font = input; startGroup; preserveAspect; ] Plot[F[x],{x,0,Sqrt[2]}]; :[font = postscript; inactive; PostScript; output; endGroup; pictureLeft = 47; pictureTop = 47; pictureWidth = 235; pictureHeight = 145; preserveAspect; ] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 7 scalefont setfont % Scaling calculations 0.0238095 0.673435 0.0147151 0.588604 [ [(0.2)] .1585 .01472 0 2 Msboxa [(0.4)] .29318 .01472 0 2 Msboxa [(0.6)] .42787 .01472 0 2 Msboxa [(0.8)] .56256 .01472 0 2 Msboxa [(1)] .69724 .01472 0 2 Msboxa [(1.2)] .83193 .01472 0 2 Msboxa [(1.4)] .96662 .01472 0 2 Msboxa [(0.2)] .01131 .13244 1 0 Msboxa [(0.4)] .01131 .25016 1 0 Msboxa [(0.6)] .01131 .36788 1 0 Msboxa [(0.8)] .01131 .4856 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .1585 .01472 m .1585 .02097 L s P [(0.2)] .1585 .01472 0 2 Mshowa p .002 w .29318 .01472 m .29318 .02097 L s P [(0.4)] .29318 .01472 0 2 Mshowa p .002 w .42787 .01472 m .42787 .02097 L s P [(0.6)] .42787 .01472 0 2 Mshowa p .002 w .56256 .01472 m .56256 .02097 L s P [(0.8)] .56256 .01472 0 2 Mshowa p .002 w .69724 .01472 m .69724 .02097 L s P [(1)] .69724 .01472 0 2 Mshowa p .002 w .83193 .01472 m .83193 .02097 L s P [(1.2)] .83193 .01472 0 2 Mshowa p .002 w .96662 .01472 m .96662 .02097 L s P [(1.4)] .96662 .01472 0 2 Mshowa p .001 w .05075 .01472 m .05075 .01847 L s P p .001 w .07768 .01472 m .07768 .01847 L s P p .001 w .10462 .01472 m .10462 .01847 L s P p .001 w .13156 .01472 m .13156 .01847 L s P p .001 w .18543 .01472 m .18543 .01847 L s P p .001 w .21237 .01472 m .21237 .01847 L s P p .001 w .23931 .01472 m .23931 .01847 L s P p .001 w .26625 .01472 m .26625 .01847 L s P p .001 w .32012 .01472 m .32012 .01847 L s P p .001 w .34706 .01472 m .34706 .01847 L s P p .001 w .374 .01472 m .374 .01847 L s P p .001 w .40093 .01472 m .40093 .01847 L s P p .001 w .45481 .01472 m .45481 .01847 L s P p .001 w .48175 .01472 m .48175 .01847 L s P p .001 w .50868 .01472 m .50868 .01847 L s P p .001 w .53562 .01472 m .53562 .01847 L s P p .001 w .58949 .01472 m .58949 .01847 L s P p .001 w .61643 .01472 m .61643 .01847 L s P p .001 w .64337 .01472 m .64337 .01847 L s P p .001 w .67031 .01472 m .67031 .01847 L s P p .001 w .72418 .01472 m .72418 .01847 L s P p .001 w .75112 .01472 m .75112 .01847 L s P p .001 w .77806 .01472 m .77806 .01847 L s P p .001 w .80499 .01472 m .80499 .01847 L s P p .001 w .85887 .01472 m .85887 .01847 L s P p .001 w .88581 .01472 m .88581 .01847 L s P p .001 w .91274 .01472 m .91274 .01847 L s P p .001 w .93968 .01472 m .93968 .01847 L s P p .001 w .99356 .01472 m .99356 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .13244 m .03006 .13244 L s P [(0.2)] .01131 .13244 1 0 Mshowa p .002 w .02381 .25016 m .03006 .25016 L s P [(0.4)] .01131 .25016 1 0 Mshowa p .002 w .02381 .36788 m .03006 .36788 L s P [(0.6)] .01131 .36788 1 0 Mshowa p .002 w .02381 .4856 m .03006 .4856 L s P [(0.8)] .01131 .4856 1 0 Mshowa p .002 w .02381 .60332 m .03006 .60332 L s P [(1)] .01131 .60332 1 0 Mshowa p .001 w .02381 .03826 m .02756 .03826 L s P p .001 w .02381 .0618 m .02756 .0618 L s P p .001 w .02381 .08535 m .02756 .08535 L s P p .001 w .02381 .10889 m .02756 .10889 L s P p .001 w .02381 .15598 m .02756 .15598 L s P p .001 w .02381 .17952 m .02756 .17952 L s P p .001 w .02381 .20307 m .02756 .20307 L s P p .001 w .02381 .22661 m .02756 .22661 L s P p .001 w .02381 .2737 m .02756 .2737 L s P p .001 w .02381 .29724 m .02756 .29724 L s P p .001 w .02381 .32079 m .02756 .32079 L s P p .001 w .02381 .34433 m .02756 .34433 L s P p .001 w .02381 .39142 m .02756 .39142 L s P p .001 w .02381 .41497 m .02756 .41497 L s P p .001 w .02381 .43851 m .02756 .43851 L s P p .001 w .02381 .46205 m .02756 .46205 L s P p .001 w .02381 .50914 m .02756 .50914 L s P p .001 w .02381 .53269 m .02756 .53269 L s P p .001 w .02381 .55623 m .02756 .55623 L s P p .001 w .02381 .57977 m .02756 .57977 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .001 w .02381 .01472 m .02505 .01472 L .02629 .01472 L .02753 .01473 L .02877 .01474 L .03125 .01477 L .03373 .01482 L .03621 .01487 L .03869 .01494 L .04365 .01512 L .04861 .01534 L .05357 .01562 L .06349 .01632 L .07341 .01722 L .08333 .01833 L .10317 .02114 L .12302 .02475 L .14286 .02916 L .18254 .0404 L .22222 .05484 L .2619 .0725 L .30159 .09337 L .34127 .11745 L .38095 .14473 L .42063 .17523 L .46032 .20894 L .5 .24586 L .53968 .28599 L .57937 .32933 L .61905 .37588 L .65873 .42564 L .69841 .47853 L .7381 .51786 L .77778 .5464 L .81746 .568 L .8373 .57663 L .85714 .58394 L .87698 .59 L .89683 .59488 L .91667 .59861 L .92659 .60006 L .93651 .60124 L .94643 .60216 L .95139 .60251 L .95635 .6028 L .96131 .60303 L .96379 .60312 L .96627 .60319 L .96875 .60325 L .96999 .60327 L Mistroke .97123 .60329 L .97247 .6033 L .97371 .60331 L .97495 .60332 L .97619 .60332 L Mfstroke P P % End of Graphics MathPictureEnd :[font = input; startGroup; preserveAspect; ] f1[x_] := Pi x^2/4 f2[x_] := Sqrt[x^2 - 1] + Pi x^2/4 - x^2 ArcCos[1/x] Integrate[1 - f1[x],{x,0,1}] + Integrate[1 - f2[x],{x,1,Sqrt[2]}] :[font = output; inactive; output; endGroup; ] 2^(1/2)/3 - (2^(3/2)*Pi)/12 + (2^(3/2)*ArcCos[2^(-1/2)])/3 + Log[1 + 2^(1/2)]/3 ;[o] 3/2 1 3/2 2 ArcCos[-------] Sqrt[2] 2 Pi Sqrt[2] Log[1 + Sqrt[2]] ------- - ------- + -------------------- + ---------------- 3 12 3 3 :[font = input; preserveAspect; ] :[font = input; startGroup; preserveAspect; ] % /. ArcCos[1/Sqrt[2]] -> Pi/4 :[font = output; inactive; output; endGroup; endGroup; ] 2^(1/2)/3 + Log[1 + 2^(1/2)]/3 ;[o] Sqrt[2] Log[1 + Sqrt[2]] ------- + ---------------- 3 3 :[font = input; startGroup; ] N[%,20] :[font = output; inactive; output; endGroup; ] 0.76519571646421269134 ;[o] 0.76519571646421269134 ^*)