<Text-field layout="Heading 1" style="Heading 1">The Midpoint Rule</Text-field>

Approximate Integration

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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 NiMvKiYlJkRlbHRhRyIiIiUieEdGJiomLCYlImJHRiYlImFHISIiRiYlIm5HRiw= . Rectangles are drawn with a base of length NiMqJiUmRGVsdGFHIiIiJSJ4R0Yl 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 NiMtJSRpbnRHNiQtJSRleHBHNiMsJCokJSJ4RyIiIyEiIi9GKzsiIiEiIiI= 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(%);

<Text-field layout="Heading 1" style="Heading 1">The Trapezoidal Rule</Text-field>

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 NiMvJSJBRyoqIiIiRiYlImhHRiYsJiUiYUdGJiUiYkdGJkYmIiIjISIi where h is the height and a and b are the two bases. In this case, NiMvJSJoRyomLCYlImJHIiIiJSJhRyEiIkYoJSJuR0Yq 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 NiMtJSRpbnRHNiQtJSRleHBHNiMsJCokJSJ4RyIiIyEiIi9GKzsiIiEiIiI= 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(%);

<Text-field layout="Heading 1" style="Heading 1">Simpson' Rule</Text-field>

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 NiMtJSJmRzYjJSJ4Rw== in the interval a < x < b and NiMvKiYlJkRlbHRhRyIiIiUieEdGJiomLCYlImJHRiYlImFHISIiRiYlIm5HRiw= , the formula to approximate the integral is as follows:

Example 3: Use Simpson's Rule to approximate NiMtJSRpbnRHNiQtJSRleHBHNiMsJCokJSJ4RyIiIyEiIi9GKzsiIiEiIiI= 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));

<Text-field layout="Heading 1" style="Heading 1">Exercises</Text-field>

<Text-field layout="Heading 1" style="Heading 1">Exercise 1: Use the sum command to approximate <Equation input-equation="int(1/sqrt(1+x^3),x = 0 .. 3);" style="_cstyle256">NiMtJSRpbnRHNiQqJiIiIkYnLSUlc3FydEc2IywmRidGJyokJSJ4RyIiJEYnISIiL0YtOyIiIUYu</Equation> with n = 20 using the midpoint rule. Confirm your answer using the middlesum command and graph the curve and the rectangles. Then compare it with the result given by the int command.</Text-field>

restart;with(student):

<Text-field layout="Normal" style="Normal">Exercise 2: Use the sum command to approximate <Equation input-equation="int(1/sqrt(1+x^3),x = 0 .. 3);" style="_cstyle256">NiMtJSRpbnRHNiQqJiIiIkYnLSUlc3FydEc2IywmRidGJyokJSJ4RyIiJEYnISIiL0YtOyIiIUYu</Equation> with n = 20 using the trapezoidal rule. Confirm your answer using the trapezoid command and compare it with the result given by the int command.</Text-field>

restart;with(student):

<Text-field layout="Heading 2" style="Heading 2">Exercise 3: Use Simpson's Rule to approximate int(sin(x^2), x = 0 .. sqrt(Pi)) with n = 20 and compare the result with those given by the midpoint comand, the trapezoid command, and the int command. Confirm that your solution is a linear combination of the midpoint rule and trapezoid rule using MAPLE.K</Text-field>

restart:with(student):

<Text-field layout="Heading 2" style="Heading 2">Exercise 4: Find the size of n necessary to approximate the integral <Equation input-equation="int(exp(-x^2), x = 3 .. 4)" style="2D Math">NiMtSSRpbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUkkZXhwR0YlNiMsJCokSSJ4R0YoIiIjISIiL0YvOyIiJCIiJQ==</Equation> to within 0.0001 of the actual integral then use the n you find to compute the value. You will want to use the error approximation given in class. </Text-field>

restart:with(student):