5.2 The Definite Integral

Mathematica script by Chris Parrish,
   cparrish@sewanee.edu

Sources and references for some of these problems include
    James Stewart, "Calculus: Concepts and Contexts," Second Edition, Brooks/Cole, 2001
    Deborah Hughes-Hallett, Andrew M. Gleason, et. al., "Calculus," Second Edition, John Wiley & Sons, 1998
    Robert Fraga, ed., "Calculus Problems for a New Century," The Mathematical Association of America, 1993
    
Code for the Mathematica procedures AreaR, AreaL, and AreaM is modified slightly from programs developed in
    Finch and Lehmann, "Exploring Calculus with
Mathematica," Addison-Wesley, 1992

AreaR, AreaL, AreaM -- contains Mathematica code only, not worked exercises

Using Rectangles to Estimate the Area of the Region
under the Graph of  f(x) = 2 - x^2and over the interval [0,2].

Stewart, Exercise 5.2.1, page 367

Clear[f, x] ; <br /> f[x_] := 2 - x^2 ;  a = 0 ;       & ... ;    (* n = number of rectangles *)AreaRVerbose[f, a, b, n] ;

[Graphics:HTMLFiles/5.2_definiteIntegral_4.gif]

RowBox[{The total area of the illustrated rectangles is , , 0.25}]

Using Rectangles to Estimate the Area of the Region
under the Graph of  f(x) = ln x - 1 and over the interval [1,4].

Stewart, Exercise 5.2.1, page 367

Clear[f,x];

f[x_] := Log[x] - 1;

a = 1;            (* a = lefthand endpoint *)
b = 4;             (* b = righthand endpoint *)
n = 6;            (* n = number of rectangles *)

AreaLVerbose[f,a,b,n];

[Graphics:HTMLFiles/5.2_definiteIntegral_6.gif]

RowBox[{The total area of the illustrated rectangles is , , RowBox[{-, 0.816861}]}]

Using Rectangles to Estimate the Area of the Region
under the Graph of  f(x) = e^(-x^2) and over the interval [0,2].

Stewart, Exercise 5.2.16, page 368

Clear[f, x] ; <br /> f[x_] := Exp[-x^2] ; <br /> a = 0 ;       & ... ;    (* n = number of rectangles *)AreaLVerbose[f, a, b, n] ;

[Graphics:HTMLFiles/5.2_definiteIntegral_10.gif]

RowBox[{The total area of the illustrated rectangles is , , 0.980007}]


Created by Mathematica  (April 6, 2004)