{"text_format":"3","workbook":"xlsx","n_sheets":2,"sheets":["z-Substitution","Sheet2"],"not_rendered":{"charts":3,"images":6,"defined_names":36,"comments":1},"read":"whole","date_system":"1900"}

{"sheet":"z-Substitution","index":0,"n_rows":54,"n_cols":19,"rows_kept":54}
[null,"       Visualizing the z-Substitution"]
[]
[null,null,"    This DIGMath spreadsheet lets you"]
[null,null,"               visualize the process of"]
[null,null,"      Integration with the z-Substitution"]
[null,null,"for rational functions of sine and cosine"]
[]
[]
[]
[]
["What is the value of the constant a?",null,null,2]
[]
["What is the value of the constant b?",null,null,3]
[{"$nulls":9},"Pick a tracing point x0 =",null,null,{"$f":"a+K15*dx","v":2.445}]
["What is the value of the constant c?",null,null,1,{"$nulls":6},163]
[]
[{"$nulls":8},"         The corresponding transformed point is z = ",{"$nulls":6},{"$f":"TAN(x0/2)","v":2.7540697327827077}]
["                The z-substitution is then  ",null,null,{"$f":"TEXT(a0/b0,\"0.00\")","v":"0.67"}]
[null,null,"                         so that   ",{"$f":"TEXT(a0/b0,\"0.00\")","v":"0.67"},{"$nulls":5}," The area from x =",null,{"$f":"a","v":0},"to x =",{"$f":"x0","v":2.445},"=",{"$f":"LOOKUP(x0+dx/2,Sheet2!$C$3:$C$203,Sheet2!E$3:E$203)","v":{"v":0.5646897585289976,"d":"0.565"}}]
[]
[]
[]
[]
[{"$nulls":15},"The area  under the transformed graph"]
[{"$nulls":15},"from z =",{"$f":"theta0","v":{"v":0,"d":"0.000"}},"to z =",{"$f":"th0","v":{"v":2.7540697327827077,"d":"2.754"}}]
[" The transformed integrand is then  ",{"$nulls":3},"2z",{"$nulls":11},"is about",{"$f":"LOOKUP(th0+dtheta/2,Sheet2!F3:F203,Sheet2!H3:H203)","v":{"v":0.5646946404448312,"d":"0.565"}}]
[{"$nulls":3},{"$f":"TEXT(a0-c0,\"0\")&\" (z^2) + \"&TEXT(2*b0,\"0\"&\"z + \"&TEXT(a0+c0,\"0\"))","v":"1 (z^2) + 6z + 3"}]
[]
[]
["What is the interval for x for display? "]
[null,null,"xMin =",0]
[null,null,"xMax =",3]
[]
[null,null,"       This corresponds to:",{"$nulls":5},"          The area  =",null,{"$f":"LOOKUP(x0+dx/2,Sheet2!$C$3:$C$203,Sheet2!F3:F203)","v":{"v":2.7540697327827,"d":"2.754"}}]
[{"$nulls":3},"z Min  =",{"$f":"theta0","v":{"v":0,"d":"0.000"}}]
[{"$nulls":3},"z Max  =",{"$f":"TAN(b/2)","v":{"v":14.101419947171719,"d":"14.101"}}]
[]
[]
["Click each item below for suggestions and investigations"]
[null,null,"   Item 1"]
[null,null,"   Item 2"]
[null,null,"   Item 3"]
[null,null,"   Item 4"]
[]
[]
[]
["To access/download all the DIGMath Apps for Calculus"]
["Click this link:",null,"DIGMath for Calculus"]
[]
["Created by:  Sheldon P. Gordon   "]
["                      Farmingdale State College    "]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Sheet2","index":1,"n_rows":403,"n_cols":15,"rows_kept":403}
[{"$nulls":6},"z-Substitution"]
[null,null,"x","f(x)","Area","z ","f(z )","Area",null,"xx","f(xx)","sweep x","sweep y","z ","f(z )"]
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[{"$f":"b","v":3},null,0.09,0.30622352956021365,0.028739708372686117,0.04503039962393285,0.6112076907382876,0.028739472656318515,null,{"$f":"J7+dx","v":0.045},0,{"$f":"IF(xx<=x0,L7+dx,L7)","v":0.045},{"$f":"K9","v":0},{"$f":"IF(xx<=x0,TAN(xx/2),N7)","v":0.022503797644024767},0]
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["dx =",null,0.12,0.29833422777316354,0.037807100843149304,0.06007210383129728,0.5945230254771546,0.03780680411965553,null,{"$f":"J9+dx","v":0.06},0,{"$f":"IF(xx<=x0,L9+dx,L9)","v":0.06},{"$f":"K11","v":0},{"$f":"IF(xx<=x0,TAN(xx/2),N9)","v":0.030009003241180714},0]
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["First graph",null,0.18000000000000005,0.28401567615702206,0.055268942951079925,0.09024378990978547,0.5634427077970618,0.055268545602057864,null,{"$f":"J13+dx","v":0.09},0,{"$f":"IF(xx<=x0,L13+dx,L13)","v":0.09},{"$f":"K15","v":0},{"$f":"IF(xx<=x0,TAN(xx/2),N13)","v":0.04503039962393285},0]
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["Second graph",null,0.2400000000000001,0.2714112351191536,0.07192430233943407,0.12057933721130536,0.5350432745918803,0.07192382858626008,null,{"$f":"J17+dx","v":0.12},0,{"$f":"IF(xx<=x0,L17+dx,L17)","v":0.12},{"$f":"K19","v":0},{"$f":"IF(xx<=x0,TAN(xx/2),N17)","v":0.06007210383129728},0]
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["Third graph",null,0.30000000000000016,0.26028807425083644,0.08786881664373687,0.15113521805829516,0.5089507692668956,0.08786828650996392,null,{"$f":"J21+dx","v":0.15000000000000002},0,{"$f":"IF(xx<=x0,L21+dx,L21)","v":0.15000000000000002},{"$f":"K23","v":0},{"$f":"IF(xx<=x0,TAN(xx/2),N21)","v":0.07514094212828505},0]
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