Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://orion.math.iastate.edu/hentzel/class.166.05 Textbook: Calculus by Varberg, Purcell, Rigdon, eight 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. January 10 6.1 Area p278:1-4, 7-10, 13-16,(22),(26),27,(28),(30),31,(32),36,37 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 273 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 ------------------------------------------------ f[x_] := x^4-2x^3+2; a = Plot[f[x],{x,-1,2}]; b = Show[a,PlotLabel->"P273ex1 x^4-2x^3+2"]; Display["x1.ps",b]; --------------------------------------------------------------- Page 274 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 ------------------------------------------------ f[x_] := x^2/3-4; a = Plot[f[x],{x,-2,3}]; b = Show[a,PlotLabel->"P274ex2 x^2/3-4"]; Display["x2.ps",b]; --------------------------------------------------------------- Page 274 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. ------------------------------------------------ f[x_] := x^3-3x^2-x+3; a = Plot[f[x],{x,-1,2}]; b = Show[a,PlotLabel->"P274ex3 x^3-3x^2-x+3"]; Display["x3.ps",b]; --------------------------------------------------------------- --------------------------------------------------------------- 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. --------------------------------------------------------------- Page 276 Example 5. Find the area of the region between the curves y = x^4 and y = 2x - x^2. Ans 7/15 ------------------------------------------------ f[x_] := x^4; g[x_] := 2x-x^2; a = Plot[{f[x],g[x]},{x,-3,3}]; b = Show[a,PlotLabel->"P276ex5 y=x^4 and y = 2x-x^2"]; Display["x5.ps",b]; --------------------------------------------------------------- --------------------------------------------------------------- Page 276 Example 6 Find the area of the region between the parabola 2 y = 4 x and the line 4x - 3y = 4. ans 125/24. ------------------------------------------------ a = ParametricPlot[{y^2/4,y},{y,-3,5}]; b = Plot[1/3 (4x-4),{x, 0,6}]; c = Show[a,b,PlotLabel->"P276ex6 y^2=4x and 4x-3y=4",AspectRatio->Automatic]; Display["x6.ps",c]; --------------------------------------------------------------- ---------------------------------------------------------------