(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; currentKernel; ] :[font = subsection; inactive; preserveAspect] Math 225: Calculus III Quiz 7 3/26&28/96 :[font = input; initialization; preserveAspect] *) <1]; :[font = input; preserveAspect; startGroup] Show[p, Graphics[{PointSize[0.02],Point[{4/(3Pi),4/(3Pi)}]}]] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 181; pictureHeight = 181] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0238095 0.952381 [ [(0.2)] .21429 .02381 0 2 Msboxa [(0.4)] .40476 .02381 0 2 Msboxa [(0.6)] .59524 .02381 0 2 Msboxa [(0.8)] .78571 .02381 0 2 Msboxa [(1)] .97619 .02381 0 2 Msboxa [(0.2)] .01131 .21429 1 0 Msboxa [(0.4)] .01131 .40476 1 0 Msboxa [(0.6)] .01131 .59524 1 0 Msboxa [(0.8)] .01131 .78571 1 0 Msboxa [(1)] .01131 .97619 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .02381 m .21429 .03006 L s P [(0.2)] .21429 .02381 0 2 Mshowa p .002 w .40476 .02381 m .40476 .03006 L s P [(0.4)] .40476 .02381 0 2 Mshowa p .002 w .59524 .02381 m .59524 .03006 L s P [(0.6)] .59524 .02381 0 2 Mshowa p .002 w .78571 .02381 m .78571 .03006 L s P [(0.8)] .78571 .02381 0 2 Mshowa p .002 w .97619 .02381 m .97619 .03006 L s P [(1)] .97619 .02381 0 2 Mshowa p .001 w .0619 .02381 m .0619 .02756 L s P p .001 w .1 .02381 m .1 .02756 L s P p .001 w .1381 .02381 m .1381 .02756 L s P p .001 w .17619 .02381 m .17619 .02756 L s P p .001 w .25238 .02381 m .25238 .02756 L s P p .001 w .29048 .02381 m .29048 .02756 L s P p .001 w .32857 .02381 m .32857 .02756 L s P p .001 w .36667 .02381 m .36667 .02756 L s P p .001 w .44286 .02381 m .44286 .02756 L s P p .001 w .48095 .02381 m .48095 .02756 L s P p .001 w .51905 .02381 m .51905 .02756 L s P p .001 w .55714 .02381 m .55714 .02756 L s P p .001 w .63333 .02381 m .63333 .02756 L s P p .001 w .67143 .02381 m .67143 .02756 L s P p .001 w .70952 .02381 m .70952 .02756 L s P p .001 w .74762 .02381 m .74762 .02756 L s P p .001 w .82381 .02381 m .82381 .02756 L s P p .001 w .8619 .02381 m .8619 .02756 L s P p .001 w .9 .02381 m .9 .02756 L s P p .001 w .9381 .02381 m .9381 .02756 L s P p .002 w 0 .02381 m 1 .02381 L s P p .002 w .02381 .21429 m .03006 .21429 L s P [(0.2)] .01131 .21429 1 0 Mshowa p .002 w .02381 .40476 m .03006 .40476 L s P [(0.4)] .01131 .40476 1 0 Mshowa p .002 w .02381 .59524 m .03006 .59524 L s P [(0.6)] .01131 .59524 1 0 Mshowa p .002 w .02381 .78571 m .03006 .78571 L s P [(0.8)] .01131 .78571 1 0 Mshowa p .002 w .02381 .97619 m .03006 .97619 L s P [(1)] .01131 .97619 1 0 Mshowa p .001 w .02381 .0619 m .02756 .0619 L s P p .001 w .02381 .1 m .02756 .1 L s P p .001 w .02381 .1381 m .02756 .1381 L s P p .001 w .02381 .17619 m .02756 .17619 L s P p .001 w .02381 .25238 m .02756 .25238 L s P p .001 w .02381 .29048 m .02756 .29048 L s P p .001 w .02381 .32857 m .02756 .32857 L s P p .001 w .02381 .36667 m .02756 .36667 L s P p .001 w .02381 .44286 m .02756 .44286 L s P p .001 w .02381 .48095 m .02756 .48095 L s P p .001 w .02381 .51905 m .02756 .51905 L s P p .001 w .02381 .55714 m .02756 .55714 L s P p .001 w .02381 .63333 m .02756 .63333 L s P p .001 w .02381 .67143 m .02756 .67143 L s P p .001 w .02381 .70952 m .02756 .70952 L s P p .001 w .02381 .74762 m .02756 .74762 L s P p .001 w .02381 .82381 m .02756 .82381 L s P p .001 w .02381 .8619 m .02756 .8619 L s P p .001 w .02381 .9 m .02756 .9 L s P p .001 w .02381 .9381 m .02756 .9381 L s P p .002 w .02381 0 m .02381 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p p .004 w .97619 .02381 m .97619 .02576 L .97618 .02771 L .97617 .02965 L .97616 .0316 L .97612 .0355 L .97606 .03939 L .97599 .04329 L .9759 .04718 L .97568 .05497 L .97539 .06276 L .97504 .07054 L .97415 .0861 L .973 .10164 L .9716 .11716 L .96804 .14812 L .96347 .17895 L .95789 .20961 L .94374 .2703 L .92565 .32994 L .90369 .38827 L .87797 .44504 L .8486 .5 L .81569 .55292 L .77938 .60358 L .73985 .65176 L .69724 .69724 L .65176 .73985 L .60358 .77938 L .55292 .81569 L .5 .8486 L .44504 .87797 L .38827 .90369 L .32994 .92565 L .2703 .94374 L .20961 .95789 L .17895 .96347 L .14812 .96804 L .11716 .9716 L .10164 .973 L .0861 .97415 L .07054 .97504 L .06276 .97539 L .05497 .97568 L .04718 .9759 L .04329 .97599 L .03939 .97606 L .0355 .97612 L .03355 .97614 L .0316 .97616 L Mistroke .02965 .97617 L .02771 .97618 L .02576 .97619 L .02381 .97619 L Mfstroke P P p .02 w .42801 .42801 Mdot P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] 2. Compute the average value of the function f[x_,y_,z_] = x^3 y over the solid region bounded in the first quadrant by the planes x + z == 2 and y == 3. The volume of the region is 6. ;[s] 7:0,0;46,1;65,0;132,1;142,0;147,1;153,0;186,-1; 2:4,13,9,Times,1,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; preserveAspect] Solution :[font = input; preserveAspect] top = Plot3D[2-x,{x,0,2},{y,0,3},PlotPoints->2]; :[font = input; preserveAspect] side = ParametricPlot3D[{x,3,s(2-x)},{x,0,2},{s,0,1}, PlotPoints->2]; :[font = input; preserveAspect; startGroup] Show[top,side,ViewPoint->{2.242, 1.966, 1.600}, BoxRatios->Automatic] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 250; pictureHeight = 230] %! %%Creator: Mathematica %%AspectRatio: .92278 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations 0.00376301 0.971939 0.0211062 0.971939 [ [(0)] .93629 .25172 -1 .43318 Msboxa [(0.5)] .86954 .20038 -1 .4548 Msboxa [(1)] .79847 .14571 -1 .47869 Msboxa [(1.5)] .72264 .08738 -1 .50522 Msboxa [(2)] .64156 .02502 -1 .53487 Msboxa [(0)] .06536 .31258 1 .5277 Msboxa [(1)] .22338 .22722 1 .58734 Msboxa [(2)] .3995 .13209 1 .66219 Msboxa [(3)] .59702 .02543 1 .75889 Msboxa [(0)] .94371 .26658 -1 .42841 Msboxa [(0.5)] .95362 .3679 -1 .4006 Msboxa [(1)] .96398 .47381 -1 .37169 Msboxa [(1.5)] .97481 .58464 -1 .34161 Msboxa [(2)] .98615 .70073 -1 .31028 Msboxa [ 0 0 0 0 ] [ 1 .92278 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .93158 .2615 m .62245 .0243 L s P p .002 w .92514 .25655 m .91957 .25897 L s P [(0)] .93629 .25172 -1 .43318 Mshowa p .002 w .85848 .20541 m .85295 .20792 L s P [(0.5)] .86954 .20038 -1 .4548 Mshowa p .002 w .78751 .15095 m .78203 .15358 L s P [(1)] .79847 .14571 -1 .47869 Mshowa p .002 w .7118 .09286 m .70638 .0956 L s P [(1.5)] .72264 .08738 -1 .50522 Mshowa p .002 w .63085 .03075 m .62549 .03361 L s P [(2)] .64156 .02502 -1 .53487 Mshowa p .001 w .91213 .24657 m .90879 .24804 L s P p .001 w .89897 .23647 m .89563 .23795 L s P p .001 w .88564 .22625 m .88231 .22773 L s P p .001 w .87215 .21589 m .86882 .21739 L s P p .001 w .84465 .19479 m .84133 .19631 L s P p .001 w .83064 .18404 m .82733 .18558 L s P p .001 w .81645 .17315 m .81315 .1747 L s P p .001 w .80207 .16213 m .79878 .16369 L s P p .001 w .77277 .13964 m .76948 .14123 L s P p .001 w .75782 .12817 m .75455 .12977 L s P p .001 w .74269 .11656 m .73942 .11817 L s P p .001 w .72735 .10479 m .72409 .10642 L s P p .001 w .69605 .08077 m .6928 .08243 L s P p .001 w .68008 .06852 m .67684 .07019 L s P p .001 w .66389 .0561 m .66066 .05779 L s P p .001 w .64749 .04351 m .64426 .04521 L s P P p p .002 w .06492 .32426 m .62245 .0243 L s P p .002 w .0761 .31825 m .08148 .32108 L s P [(0)] .06536 .31258 1 .5277 Mshowa p .002 w .23385 .23337 m .23909 .23645 L s P [(1)] .22338 .22722 1 .58734 Mshowa p .002 w .40963 .1388 m .41469 .14216 L s P [(2)] .3995 .13209 1 .66219 Mshowa p .002 w .6067 .03277 m .61154 .03644 L s P [(3)] .59702 .02543 1 .75889 Mshowa p .001 w .10635 .30198 m .10956 .3037 L s P p .001 w .13722 .28536 m .14042 .28712 L s P p .001 w .16875 .2684 m .17193 .27019 L s P p .001 w .20096 .25107 m .20412 .25289 L s P p .001 w .26747 .21529 m .2706 .21716 L s P p .001 w .30183 .1968 m .30494 .19871 L s P p .001 w .33696 .1779 m .34005 .17984 L s P p .001 w .37288 .15857 m .37595 .16055 L s P p .001 w .44722 .11858 m .45023 .12062 L s P p .001 w .48569 .09788 m .48868 .09996 L s P p .001 w .52507 .07669 m .52803 .07881 L s P p .001 w .5654 .05499 m .56833 .05716 L s P P p p .002 w .93158 .2615 m .9757 .71621 L s P p .002 w .93254 .27136 m .92696 .27375 L s P [(0)] .94371 .26658 -1 .42841 Mshowa p .002 w .94234 .37241 m .93671 .37467 L s P [(0.5)] .95362 .3679 -1 .4006 Mshowa p .002 w .95259 .47804 m .9469 .48016 L s P [(1)] .96398 .47381 -1 .37169 Mshowa p .002 w .96332 .58856 m .95757 .59053 L s P [(1.5)] .97481 .58464 -1 .34161 Mshowa p .002 w .97455 .70433 m .96875 .70613 L s P [(2)] .98615 .70073 -1 .31028 Mshowa p .001 w .93447 .29122 m .93111 .29264 L s P p .001 w .93641 .31125 m .93305 .31266 L s P p .001 w .93837 .33146 m .935 .33285 L s P p .001 w .94035 .35185 m .93697 .35322 L s P p .001 w .94436 .39316 m .94097 .3945 L s P p .001 w .94639 .4141 m .94299 .41542 L s P p .001 w .94844 .43522 m .94504 .43653 L s P p .001 w .95051 .45653 m .9471 .45782 L s P p .001 w .9547 .49974 m .95128 .501 L s P p .001 w .95682 .52164 m .95339 .52288 L s P p .001 w .95897 .54375 m .95553 .54496 L s P p .001 w .96113 .56605 m .95769 .56725 L s P p .001 w .96552 .61128 m .96207 .61244 L s P p .001 w .96775 .63422 m .96428 .63536 L s P p .001 w .96999 .65737 m .96652 .65849 L s P p .001 w .97226 .68074 m .96879 .68184 L s P P 0 0 m 1 0 L 1 .92278 L 0 .92278 L closepath clip newpath p .002 w .3905 .49477 m .38214 .89848 L s .38214 .89848 m .9757 .71621 L s .9757 .71621 m .93158 .2615 L s .93158 .2615 m .3905 .49477 L s .06492 .32426 m .62245 .0243 L s .62245 .0243 m .63785 .52544 L s .63785 .52544 m .0243 .76576 L s .0243 .76576 m .06492 .32426 L s .3905 .49477 m .38214 .89848 L s .38214 .89848 m .0243 .76576 L s .0243 .76576 m .06492 .32426 L s .06492 .32426 m .3905 .49477 L s .93158 .2615 m .62245 .0243 L s .62245 .0243 m .63785 .52544 L s .63785 .52544 m .9757 .71621 L s .9757 .71621 m .93158 .2615 L s P p .002 w .95098 .70557 m .3868 .88153 L .08394 .33245 L p .572 .544 .785 r F P s P p .002 w .08394 .33245 m .61543 .04983 L .95098 .70557 L p .572 .544 .785 r F P s P p .002 w .91105 .27291 m .61543 .04983 L .95098 .70557 L closepath p .934 .656 .491 r F P s P p .002 w .06492 .32426 m .62245 .0243 L s .62245 .0243 m .63785 .52544 L s .63785 .52544 m .0243 .76576 L s .0243 .76576 m .06492 .32426 L s .93158 .2615 m .62245 .0243 L s .62245 .0243 m .63785 .52544 L s .63785 .52544 m .9757 .71621 L s .9757 .71621 m .93158 .2615 L s P p p .002 w .93158 .2615 m .62245 .0243 L s P p .002 w .92514 .25655 m .91957 .25897 L s P [(0)] .93629 .25172 -1 .43318 Mshowa p .002 w .85848 .20541 m .85295 .20792 L s P [(0.5)] .86954 .20038 -1 .4548 Mshowa p .002 w .78751 .15095 m .78203 .15358 L s P [(1)] .79847 .14571 -1 .47869 Mshowa p .002 w .7118 .09286 m .70638 .0956 L s P [(1.5)] .72264 .08738 -1 .50522 Mshowa p .002 w .63085 .03075 m .62549 .03361 L s P [(2)] .64156 .02502 -1 .53487 Mshowa p .001 w .91213 .24657 m .90879 .24804 L s P p .001 w .89897 .23647 m .89563 .23795 L s P p .001 w .88564 .22625 m .88231 .22773 L s P p .001 w .87215 .21589 m .86882 .21739 L s P p .001 w .84465 .19479 m .84133 .19631 L s P p .001 w .83064 .18404 m .82733 .18558 L s P p .001 w .81645 .17315 m .81315 .1747 L s P p .001 w .80207 .16213 m .79878 .16369 L s P p .001 w .77277 .13964 m .76948 .14123 L s P p .001 w .75782 .12817 m .75455 .12977 L s P p .001 w .74269 .11656 m .73942 .11817 L s P p .001 w .72735 .10479 m .72409 .10642 L s P p .001 w .69605 .08077 m .6928 .08243 L s P p .001 w .68008 .06852 m .67684 .07019 L s P p .001 w .66389 .0561 m .66066 .05779 L s P p .001 w .64749 .04351 m .64426 .04521 L s P P p P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics3D["<<>>"] ;[o] -Graphics3D- :[font = text; inactive; preserveAspect] The region is defined by 0 <= z <= 2 - x 0 <= y <= 3 0 <= x <= 2 The volume is obviously 1/2 the volume of the box: 3*2*2 / 2 = 6. ;[s] 3:0,0;25,1;77,0;143,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] v = Integrate[1,{x,0,2},{y,0,3},{z,0,2-x}] :[font = output; output; inactive; preserveAspect; endGroup] 6 ;[o] 6 :[font = text; inactive; preserveAspect] The average value of f[x_,y_] = x^3 y is given by the integral of f[x,y] over the region divided by the volume of the region v. ;[s] 7:0,0;21,1;37,0;66,1;72,0;125,1;126,0;128,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] fbar = Integrate[x^3 y,{x,0,2},{y,0,3},{z,0,2-x}]/v :[font = text; inactive; preserveAspect] One step at a time: :[font = input; preserveAspect; startGroup] Integrate[x^3 y,{z,0,2-x}] :[font = output; output; inactive; preserveAspect; endGroup] (2 - x)*x^3*y ;[o] 3 (2 - x) x y :[font = input; preserveAspect; startGroup] Integrate[%,{y,0,3}] :[font = output; output; inactive; preserveAspect; endGroup] (9*(2 - x)*x^3)/2 ;[o] 3 9 (2 - x) x ------------ 2 :[font = input; preserveAspect; startGroup] Integrate[%,{x,0,2}] :[font = output; output; inactive; preserveAspect; endGroup] 36/5 ;[o] 36 -- 5 :[font = input; preserveAspect; startGroup] fbar = %/v :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 6/5 ;[o] 6 - 5 ^*)