Some key commands are:
| leftbox | rightbox | middlebox | |
| leftsum | rightsum | middlesum | |
| trapezoid | simpson | ||
| value | evalf |
NOTE: First you must load the necessary commands by entering:
> with( student ) ;
I. Graphing Regions & Rectangles
Example: Suppose we want to graph the region bound between the function
the x axis, and the vertical lines x = –1, x = 2, and show rectangles.
> f := 4 - x^2 ;
defines function f as an expression
> rightbox( f , x = -1.0 .. 2.0, 6 ) ;
plots f showing 6 right–endpoint rectangles
> leftbox( f , x = -1.0 .. 2.0 , 12 ) ;
plots f showing 12 left–endpoint rectangles
> middlebox( f , x = -1.0 .. 2.0 , 10 ) ;
plots f showing 10 midpoint rectangles
II. Approximating Areas of Regions Using Rectangles
Example: Suppose we want to approximate the area of the region bound between the function
the x axis, and the vertical lines x = –1 and x = 2.
> f := 4 - x^2 ;
defines function f as an expression
> R := rightsum( f , x = -1.0 .. 2.0 , 6 ) ;
using 6 right–endpoint rectangles
> value( R ) ;
numerical value of the result R
> evalf( R ) ;
decimal value of the result R
> L := leftsum( f , x = -1.0 .. 2.0 , 12 ) ;
using 12 left–endpoint rectangles
> value( L ) ;
numerical value of the result L
> evalf( L ) ;
decimal value of the result L
> M := middlesum( f , x = -1.0 .. 2.0 , 10 ) ;
using 10 midpoint rectangles
> evalf( M ) ;
decimal value of the result M
III. Approximating Areas / Integrals Using the Trapezoidal Rule
Example: Suppose we want to approximate the area of the region bound between the function
the x axis, and the vertical lines x = –1 and x = 2.
> f := exp( -x^2 ) ;
defines function f as an expression
> T := trapezoid( f , x = -1.0 .. 2.0 , 6 ) ;
using 6 subintervals
> evalf( T ) ;
decimal value of the result T
IV. Approximating Areas / Integrals Using Simpson's (1/3) Rule
Note: &nsbp; The number of subintervals must be even.
Example: Suppose we want to approximate the area of the region bound between the function
the x axis, and the vertical lines x = –1 and x = 2.
> f := exp( -x^2 ) ;
defines function f as an expression
> S := simpson( f , x = -1.0 .. 2.0 , 6 ) ;
using 6 subintervals
> evalf( S ) ;
decimal value of the result S
V. Exact Areas Using Limits
Example: Determine the exact area of the region bound between the function
the x axis, and the vertical lines x = –1 and x = 2.
> n:='n';
resets n in case it was previously assigned a value
> f := 4 - x^2 ;
defines function f as an expression
> approx := rightsum( f , x = -1.0 .. 2.0 , n ) ;
uses n right–endpoint rectangles
> Riemann := value( approx ) ;
simplifies the sums and calls result Riemann
> area := limit( Riemann , n = infinity ) ;
lets the number of rectangles go to infinity
NOTE: For many other commands common to student use, enter
> ?student;
Written and Maintained by
Last modified: 03/28/2020
Copyright © 1997–2023 Kevin G. TeBeest. All rights reserved.
Maple® is a registered trademark of Waterloo Maple Software.
Prof. Kevin G. TeBeest
Applied Mathematics
Kettering University