Approximate Integration Name: Approximation methods for integrals are necessary since not all functions have antiderivatives. Three methods will be investigated: the Midpoint Rule, the Trapezoidal Rule, and Simpson's Rule.

An interval a < x < b is divided into n subintervals, each of length 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 . Rectangles are drawn with a base of length 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 and height the function value taken at the midpoint of the interval. To obtain better estimates, the number of rectangles can be increased. Example 1: Use the sum function to find the midpoint approximation to 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 with n = 10. f:=x->exp(-x^2); s:=sum(1/10*f(.5*(i/10+(i+1)/10)),i=0..9); The answer can be confirmed and a plot drawn using commands contained in the student package. with(student): middlebox(f(x),x=0..1,10); evalf(middlesum(f(x),x=0..1,10)); Compare the result with what is given by the int command. int(f(x),x=0..1); evalf(%);

The length of each subinterval is found as above, but instead of rectangles, trapezoids are inscribed in each subinterval. The following sequence of commands will plot a curve and the inscribed trapezoids. The formula for the area of a trapezoid is 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 where h is the height and a and b are the two bases. In this case, 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 and a and b are the values of the function, evaluated at the two endpoints of each interval. The trapezoidal approximation is then given by: restart; with(plots):with(student): Input the function, change it to any one you wish f:=x->sqrt(x): Input the limits of integration of your choice a:=0:b:=6: The next statement will plot the function in the display command below p:=plot(f(x),x=a..b,thickness=3,color=black): Choose the number of subintervals n:=6: This statement creates an array for the trapezoids c:=array(0..n): The for do loop creates the trapezoids with random colors for i from 0 to n-1 do c[i]:=polygonplot([[a+i*(b-a)/n,0],[a+(i+1)*(b-a)/n,0],[a+(i+1)*(b-a)/n,f(a+(i+1)*(b-a)/n)],[a+i*(b-a)/n,f(a+i*(b-a)/n)]],color=COLOR(RGB,rand()/10^12,rand()/10^12,rand()/10^12)):od: This command displays the function and the inscribed trapezoids display({seq(c[i],i=0..n-1),p}); Example 2: Use the sum command to find the trapezoidal approximation to 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 with n = 20, compare it with the result given by the trapezoid and the int commands. restart;with(student): f:=x->exp(-x^2): s:=sum((1/20)/2*(f(i/20)+f((i+1)/20)),i=0..19); evalf(%); trapezoid(f(x),x=0..1,20); evalf(%); int(f(x),x=0..1); evalf(%);

Simpson's Rule is based on the use of parabolas to approximate a curve. If n (where n is even) subintervals are used for a given function 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 in the interval a < x < b and 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 , the formula to approximate the integral is as follows: Example 3: Use Simpson's Rule to approximate 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with n = 20 and compare the result with those given by the midpoint command, the trapezoid command, and the int command. restart;with(student): f:=x->exp(-x^2): simpson(f(x),x=0..1,20); evalf(%); evalf(trapezoid(f(x),x=0..1,20)); evalf(middlesum(f(x),x=0..1,20)); evalf(int(f(x),x=0..1));

restart:with(student):

restart;with(student):

restart;with(student):