Math 166 Calculus II 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 Textbook: Calculus by Varberg, Purcell, Rigdon, ninth edition. Include on your answer sheet 1. Your name. 2. Your School's name. 3. Your School's Fax Number 4. The Name of the Site Supervisor or the contact person 5. The Phone number of the Site Supervisor or the contact person. 6. Your School's address. Monday, January 14 5.1 Area p280:20,24,29,30 Main Idea: Integrate the function to find the area under the curve. Key Words: Integration, Area, Distance, Riemann Sums, Cross Section. Goal: Review area. Page 275 Example 1: Find the area of the region R under 4 3 y = x - 2 x + 2 between x = -1 and x = 2. x= 2 4 3 | 5 4 |x=2 INT x - 2 x + 2 dx = | x /5 - 2 x /4 + 2x | x=-1 | |x=-1 = 32/5 -32/4 + 4 - (-1/5 -2/4 -2) = 33/5 -30/4 + 6 = (66 -75 + 60)/10 = 51/10 -------------------------------------- Get["font.math"]; f[x_] := x^4 - 2 x^3 + 2; a = Plot[f[x],{x,-1,2}]; b = ParametricPlot[{-1,y},{y,0,f[-1]}]; c = ParametricPlot[{ 2,y},{y,0,f[ 2]}]; d = Show[a,b,c,PlotLabel->"P275 Ex 1: y=x^4-2x^3+2",AspectRatio->Automatic]; Display["ex1x.ps",d]; e = ParametricPlot[ t{-1, 3 } + (1-t) {2,0},{t,0,1}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; g = Show[a,b,c,e,PlotLabel->"P275 Ex 1: y=x^4-2x^3+2", AspectRatio->Automatic]; Display["ex1y.ps",g]; ------------------------------------------ Page 275 Example 2. Find the area for the region R bounded by 2 y = x /3 -4, the x-axis, x = -2, and x = 3 x=3 2 | 3 |x=3 8-153 INT x /3 -4 dx = | 1/3 x /3 - 4x | = 3-12 -(-8/9+8) = 8/9-17 = ----- x=-2 | |x=-2 9 = -145/9 = -16.1111 Approximation: 1/2( f[-2]+f[0])*2 + 1/2( f[0]+f[3])*3 = -85/6 = -14.1667 -------------------------------------- Get["font.math"]; f[x_] := x^2/3-4; a = Plot[f[x],{x,-2,3}]; b = ParametricPlot[{-2,y},{y,0,f[-2]}]; c = ParametricPlot[{ 3,y},{y,0,f[ 3]}]; d = Show[a,b,c,PlotLabel->"P275 Ex 2: y= x^2/3-4",AspectRatio->Automatic]; Display["ex2x.ps",d]; e = ParametricPlot[ x{-2,f[-2]}+(1-x){0,f[0]},{x,0,1}, PlotStyle->{RGBColor[1,0,0]}]; g = ParametricPlot[ x{ 3,f[ 3]}+(1-x){0,f[0]},{x,0,1}, PlotStyle->{RGBColor[1,0,0]}]; h = Show[a,b,c,e,g,PlotLabel->"P275 Ex 2: y= x^2/3-4", AspectRatio->Automatic]; Display["ex2y.ps",h]; ------------------------------------------ Page 276 Example 3. Find the area of the region R bounded by 3 2 y = x - 3x - x + 3, the segment of the x-axis between x=-1 and x=2, and the line x=2. x= 1 3 2 | 4 3 2 | x=1 INT x - 3 x - x + 3 = | x /4 - 3 x /3 -x /2 + 3x | x=-1 | | x=-1 = 1/4 -1 -1/2 +3 -( 1/4 + 1 -1/2 -3) = 4 x=2 3 2 | 4 3 2 | x=2 INT x - 3 x - x + 3 = | x /4 - 3 x /3 -x /2 + 3x | x=1 | | x=1 4-8-2+6-(1/4 - 1 -1/2 +3 ) = -7/4 The "area" is 4+7/4 = 23/4 = 5.75 -------------------------------------- Get["font.math"]; f[x_] := x^3 - 3x^2 - x + 3; a = Plot[f[x],{x,-1,2}]; b = ParametricPlot[{ 2,y},{y,0,f[ 2]}]; c = Show[a,b,PlotLabel->"P276 Ex 3: y= x^3 - 3x^2 -x + 3", AspectRatio->Automatic]; Display["ex3.ps",c]; ------------------------------------------ Page 277 Example 5. Find the area of the region between the 4 2 curves y = x and y = 2x - x . Ans 7/15 4 2 x = 2x - x 4 2 x + x -2x = 0 2 x(x-1)(x + x + 2) = 0 -1 +/- Sqrt[1-8] x=0, x=1, x = ----------------- 2 x=1 2 4 | 2 3 5 | x=1 INT 2x-x -x = | 2x /2 - x /3 -x /5 | = 1-1/3 -1/5 -(0) = 7/15 x=0 | | x=0 -------------------------------------- Get["font.math"]; f[x_] := x^4; g[x_] := 2x - x^2; a = Plot[f[x],{x,-1,2}]; b = Plot[g[x],{x,-1,2}]; c = Show[a,b,PlotLabel->"P277 Ex 5: f[x]=x^4, g[x]=2x-x^2 ", AspectRatio->Automatic,PlotRange->{{-0.5,1.5},{-0.5,1.5}}]; Display["ex5.ps",c]; ------------------------------------------ Page 278 Example 6 Find the area of the region between the parabola 2 y = 4 x and the line 4x - 3y = 4. ans 125/24. 2 x = y /4 = (4+3y)/4 2 y - 3y -4 = 0 (y+1)(y-4) = 0 y=-1, y=4 y= 4 2 INT (4+3y)/4 - y /4 dy = y=-1 y= 4 | 2 3 | y= 4 1/4 INT 4+3y-y^2 dy = 1/4 | 4y + 3y /2 - y /3 | y=-1 | | y=-1 = 1/4( 16 + 48/2 - 64/3 - (-4 + 3/2 + 1/3)) = 1/4( 40 -64/3 - (-4 + 3/2 +1/3) ) = 1/4( 40 - 64/3 + 4 - 3/2 - 1/3 ) = 1/4( 44 - 3/2 - 65/3 ) = 1/4 (264 - 9 -130)/6 = 125/24 = 5.20833 -------------------------------------- Get["font.math"]; g[x_] := (4x-4)/3; p1 = ParametricPlot[{y^2/4,y},{y,-1.5,4.5}]; p2 = Plot[g[x],{x, 0,5}]; p3 = Show[p1,p2,PlotLabel->"P278 Ex 6: y^2=4x, 4x-3y = 4 ", AspectRatio->Automatic]; Display["ex6x.ps",p3]; count = 10; line[y_] := Graphics[{RGBColor[1,0,0],Thickness[0.001], Line[{ { y^2/4, y-0.5/count}, {(4+3y)/4, y-0.5/count}, {(4+3y)/4, y+0.5/count}, {y^2/4, y+0.5/count}, {y^2/4, y-0.5/count}}]}]; p4 = Table[ line[y],{y,-1+1/count,4-1/count,1/count}]; p5 = Show[p1,p2,p4,PlotLabel->"P278 Ex 6: y^2=4x, 4x-3y = 4 ", AspectRatio->Automatic]; Display["ex6y.ps",p5]; ------------------------------------------