Integration - Area between 2 curves (1 Viewer)

kevda1st · Feb 5, 2009

Im getting these wrong, and cant find where i made a mistake.

1.Tangents at (4,2) and (-4,2) from the function X<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 12"><meta name="Originator" content="Microsoft Word 12"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CKevin%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_filelist.xml"><link rel="themeData" href="file:///C:%5CDOCUME%7E1%5CKevin%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_themedata.thmx"><link rel="colorSchemeMapping" href="file:///C:%5CDOCUME%7E1%5CKevin%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_colorschememapping.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w

oNotPromoteQF/> <w:LidThemeOther>EN-AU</w:LidThemeOther> <w:LidThemeAsian>KO</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w

ontVertAlignCellWithSp/> <w

ontBreakConstrainedForcedTables/> <w

ontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> <w:UseFELayout/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="--"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><style> <!-- /* Font Definitions */ @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:1; mso-generic-font-family:roman; mso-font-format

ther; mso-font-pitch:variable; mso-font-signature:0 0 0 0 0 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; mso-fareast-language:EN-AU;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-size:10.0pt; mso-ansi-font-size:10.0pt; mso-bidi-font-size:10.0pt;} @page Section1 {size:612.0pt 792.0pt; margin:3.0cm 72.0pt 72.0pt 72.0pt; mso-header-margin:36.0pt; mso-footer-margin:36.0pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> </style><sup>[FONT=&quot]2[/FONT]</sup> = 8Y are drawn and itesect on the y axis. Find the area between the curve and the tangents.

2. Find area beween y= root(3-x) and 2x +3y -6 =0

3. Find area between (x+1)(x-1)(x-3)=0 and (x-1)(x+1)=0

4.Find area between y=x<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 12"><meta name="Originator" content="Microsoft Word 12"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CKevin%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_filelist.xml"><link rel="themeData" href="file:///C:%5CDOCUME%7E1%5CKevin%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_themedata.thmx"><link rel="colorSchemeMapping" href="file:///C:%5CDOCUME%7E1%5CKevin%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_colorschememapping.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w

oNotPromoteQF/> <w:LidThemeOther>EN-AU</w:LidThemeOther> <w:LidThemeAsian>KO</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w

ontVertAlignCellWithSp/> <w

ontBreakConstrainedForcedTables/> <w

ontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> <w:UseFELayout/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="--"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><style> <!-- /* Font Definitions */ @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:1; mso-generic-font-family:roman; mso-font-format

ther; mso-font-pitch:variable; mso-font-signature:0 0 0 0 0 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; mso-fareast-language:EN-AU;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-size:10.0pt; mso-ansi-font-size:10.0pt; mso-bidi-font-size:10.0pt;} @page Section1 {size:612.0pt 792.0pt; margin:3.0cm 72.0pt 72.0pt 72.0pt; mso-header-margin:36.0pt; mso-footer-margin:36.0pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> </style><sup>[FONT=&quot]2[/FONT]</sup>(1-x) and y=x(1-x)<meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 12"><meta name="Originator" content="Microsoft Word 12"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CKevin%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_filelist.xml"><link rel="themeData" href="file:///C:%5CDOCUME%7E1%5CKevin%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_themedata.thmx"><link rel="colorSchemeMapping" href="file:///C:%5CDOCUME%7E1%5CKevin%5CLOCALS%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_colorschememapping.xml"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w

unctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w

oNotPromoteQF/> <w:LidThemeOther>EN-AU</w:LidThemeOther> <w:LidThemeAsian>KO</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w

ontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w

ontVertAlignCellWithSp/> <w

ontBreakConstrainedForcedTables/> <w

ontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> <w:UseFELayout/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="--"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><style> <!-- /* Font Definitions */ @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:1; mso-generic-font-family:roman; mso-font-format

ther; mso-font-pitch:variable; mso-font-signature:0 0 0 0 0 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman"; mso-fareast-language:EN-AU;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-size:10.0pt; mso-ansi-font-size:10.0pt; mso-bidi-font-size:10.0pt;} @page Section1 {size:612.0pt 792.0pt; margin:3.0cm 72.0pt 72.0pt 72.0pt; mso-header-margin:36.0pt; mso-footer-margin:36.0pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> </style><sup>[FONT=&quot]2

Thanks in advance.
[/FONT]</sup><sup>[FONT=&quot][/FONT]</sup>

independantz · Feb 5, 2009

1.
$y=x^2/8\\\therefore dy/dx=x/4\\\text{ when: }x=4,dy/dx=-1\\\text{ When: } x=-4,dy/dx=-1\\\text{ The general equation of a line is : } y-y_{1}=m[x-x_{1}]\\ \therefore \text{tangent at : } (4,2)\text{is } y-2=1(x-4) =>y=x-2\\\text{similarly,second tangent is : } y=-2-x\\\text{ Now, when }x=o , y=2\therefore\text{ the tangents intersect at: } (0,2)\\\text{ Now, If you draw this curve you can split the area into a triangle and the area under the curve and above the x-axis.}\\\text{For the area of the triangle, we need to find the x-intercepts of the tangents, which are found by subbing} y=o \text{ into their respective equations, they are: } (2,0),(-2,0), \text{ the area of the triangle is: } 4 units^2\\\text{ now for the area under the curve: }\int_{-4}^{4}x^2/8.dx\\=[x^3/24]\binom{4}{-4}=16/3 units^2\\\therefore \text{total area is } 28/3 units^2$

2.
$y=\sqrt{3-x} \text{-(1) and } 3y+2x=6\text{-(2)}\\\text{From, eq 1 and eq 2 we get: } 3\sqrt{3-x}+2x=6\\\therefore 3.\sqrt{3-x}=2(-x+3)\text{ ,Now by squaring both sides: }\\9(3-x)=4(3-x)^2\\=>4(3-x)^2-9(3-x)\\=>(3-x)(4(3-x)-9)=0\\\therefore (3-x)(3-4x)=0\\\therfore \text{ the equations intersect when x= 3 and x=3/4}\\ \text{ you should draw a graph and you will notice the the line is underneath the square rooted graph } \\\therefore \text{to find the area betweeen the curves you simply have to subtract the area of the higher graph from the lower graph, so the difference is calculated. }\\i.e.\int_{3/4}^{3}[\sqrt{3-x}-(-2x/3+2)].dx=\\\int_{3/4}^{3}(3-x)^{1/2}.dx+\int_{3/4}^{3}[2x/3-2].dx=\\ ([3-x]^{3/2}/(-3/2))^{3}_{3/4}+[x^2/3]^{3}_{3/4}-[2x]^{3}_{3/4}=\\0+9/4+9/3-3/16-6+3/2\\=9/16 units^2$

3.
$\text{ consider the areas separately and add them later, now, the area between the curve (x+1)(x-1) and the x-axis , can be found through integration : }\\\left |\int_{-1}^{1}(x^2-1).dx \right |\text{, the absolute value is required because we are finding and area and this integral will give a negative value as the area we need is under the x-axis.}\\=\left |[x^3/3-x]^{1}_{-1} \right |\\=\left |(1/3)-1-(-1/3+1) \right |\\=\left |-4/3 \right |\\=4/3 units^2\\\text{Now, for the other graph: } \int_{-1}^{1}(x^2-1)(x-3)\\=\int_{-1}^{1}[x^3-x-3x^2+3].dx\\=[x^4/4-x^2/2-x^3+3x]^{1}_{-1}\\=1/4-1/2-1+3[1/4-1/2+1-3]\\=4units^2\\\therefore \text{total area :}16/3 units^2$

Integration - Area between 2 curves (1 Viewer)

kevda1st

Member

independantz

Member

Users Who Are Viewing This Thread (Users: 0, Guests: 1)