Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.08 Office Hours 9:15-10:15 MTWThF Textbook: Calculus by Varberg, Purcell, Rigdon, eight edition. February 7 7.4 Rat'n Subst's p403: 1,(4),6,9-13,(18),21,27,29,(32),(33) Main Idea: Draw a triangle. Sines and Cosines are your friends. Much better than radicals. Key Words: Triangular substitution. Goal: Learn to integrate semi-complicated things with Square roots in them. -------------------------------------------------------- Previous assignment: p399: (6),(12),(24),(32) Page 399 Problem 6 x=Pi/2 6 INT Sin[x] dx x=0 3 x=Pi/2 | 1-Cos[2x] | INT | ------------| dx x=0 | 2 | x=Pi/2 3 1/8 INT (1-Cos[2x]) dx x=0 x=Pi/2 1/8 INT 1 - 3 Cos[2x] + 3 Cos[2x]^2 - Cos[2x]^3 dx x=0 x=Pi/2 1+Cos[4x] 1/8 INT 1 - 3 Cos[2x] + 3 ---------- - (1-Sin[2x]^2 Cos[2x] dt x=0 2 | | x=Pi/2 1/8 | x - 3 Sin[2x]/2 + 3/2 x + 3/2 Sin[4t]/4 - (1-Sin[2x])^3 /(-6) | | | x=0 1/8 ( Pi/2 - 0 + 3Pi/4 + 0 +1/6 +(-1/6)) = 5 Pi/32 --------------------------------------------------------------- Integrate[Sin[x]^6,{x,0,Pi/2}] -------------------------------------------------------------------- Page 399 Problem 12 6 2 INT Cos[x] Sin[x] dx INT Cos[x]^4 (Cos[x] Sin[x])^2 dx 2 2 |1+Cos[2x] | | Sin[2x] | INT |----------| |---------| dx | 2 | | 2 | 2 2 1/16 INT (1+Cos[2x] ) Sin[2x] dx 1/16 INT ( 1 + 2 Cos[2x] + Cos[2x]^2) Sin[2x]^2 dx 1/16 INT ( 1 + 2 Cos[2x]) Sin[2x]^2 dx + 1/16 INT Cos[2x]^2) Sin[2x]^2 dx 1-Cos[4x] 2 1/16 INT ----------- + 2 Sin[2x]^2 Cos[2x] dx + 1/16 INT 1/4 Sin[4x] dx 2 1/16 (x/2 -Sin[4x]/8 + 2 Sin[2x]^3/6 + 1/64 INT 1/2 (1-Cos[8x]) dx 1/16 (x/2 -Sin[4x]/8 + 2 Sin[2x]^3/6) + 1/128 (x - Sin[8x]/8 ) + C --------------------------------------------------------------------------- Page 399 Problem 24 5 INT Ctn [2t] dt ------------------------------------------------------------------------------ Page 399 Problem 32 ========================================================== New Material: Triangular substitution. Example 4 Page 382 INT Sqrt[a^2 - x^2] dx ans: a^2/2 ArcSin[x/a] + x/2 Sqrt[a^2 - x^2] + C Example 5 Page 383 dx INT ----------- Sqrt[9+x^2] | | ans: ln | Sqrt[9+x^2] + x | + C | | Example 6 Page 383 x=4 Sqrt[x^2-4] INT ------------ dx x=2 x ans: 2 Sqrt[3] - 2 Pi/3 = 1.36971 -------------------------------------------------------------------- f[x_] := Sqrt[x^2-4]/x; a = Plot[f[x],{x,2,5}]; b = ParametricPlot[{4,y},{y,0,f[4]},PlotStyle->{RGBColor[1,0,0]}]; c = Show[a,b,PlotLabel->"P383 ex6: y=Sqrt[x^2-4]/x",AspectRatio->Automatic]; Display["ex6.ps",c]; d = Table[ ParametricPlot[{i/4,y},{y,0,f[i/4]},PlotStyle->{RGBColor[1,0,0]}],{i,9,16}]; e = Table[ Plot[f[i/4],{x,i/4,(i+1)/4},PlotStyle->{RGBColor[1,0,0]}],{i,9,15}]; h = Show[a,b,d,e,PlotLabel->"P383 ex6: y=Sqrt[x^2-4]/x",AspectRatio->Automatic]; Display["ex6x.ps",h]; uppersum = Sum[ f[i/4],{i,9,16}] 1/4; lowersum = Sum[ f[i/4],{i,8,15}] 1/4; ------------------------------------------------------------------------- Lower Sum = 1.23615 < 1.36971 < 1.45266 = Upper Sum Example 7 Page 384 dx INT --------------------- 2 Sqrt[x + 2x + 26] ans: 2 Sqrt[x^2 + 2x + 26] - 2 ln | Sqrt[x^2 + 2x + 26] + x + 1] + C ###################################################################### Rationalizing substitution: dx INT -------- x-Sqrt[x] 1/3 INT x (x-4) dx