Some key commands are:
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leftbox |
rightbox |
middlebox |
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leftsum |
rightsum |
middlesum |
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trapezoid |
simpson |
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value |
evalf |
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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
f (x) = 4 - x2,
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..2,6); plots f showing 6 right-ended rectangles
> leftbox(f,x=-1..2,12); plots f showing 12 left-ended rectangles
> middlebox(f,x=-1..2,10); 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
f (x) = 4 - x2,
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..2,6); using 6 right-ended rectangles
> value(R); numerical value of the result R
> evalf(R); decimal value of the result R
> L:= leftsum(f,x=-1..2,12); using 12 left-ended rectangles
> value(L); numerical value of the result L
> evalf(L); decimal value of the result L
> M:= middlesum(f,x=-1..2,10); 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
f (x) = e-x2,
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..2,6); using 6 subintervals
> evalf(T); decimal value of the result T
IV. Approximating Areas / Integrals Using Simpson's (1/3) Rule
Note: The number of subintervals must be even.
Example: Suppose we want to approximate the area of the region bound between the function
f (x) = e-x2,
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..2,6); 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
f (x) = 4 - x2,
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..2,n); uses n right-sided rectangles
> simp:= value(approx); simplifies the sums and calls result simp
> area:=limit(simp,n=infinity); lets the number of rectangles go to infinity
NOTE: For many other commands common to student use, enter
> ?student;
