MAT 156 F'09 Lab 5 Midpoint and Trapezoid Rules Topic 1 : Midpoint Rule In addition to the Left Hand Rule and Right Hand Rules for calculating Riemann sums there is also the Midpoint Rule. For this rule one simply picks the middle point of each subinterval instead of one of the endpoints. The Maple commands for this are very similar to the ones for left and right hand endpoints. with(student): with(plots):

Warning, the name changecoords has been redefined

f:=x->sqrt(x);

middlebox(f(x), x=1..4, 5);

evalf(middlesum(f(x), x=1..4, 5)); Assignment Part 1: Write commands that show the boxes associated to the midpoint rule with 10, 20, 40 subintervals. Write commands that evaluate the midpoint rule with 10, 20, 40 subintervals. Assignment Part 2: Write commands that show the boxes associated to the left-hand sums, the right-hand sums and the midpoint sums with 10, 20, 40 subintervals. Do not forget to introduce the plots package. What do you notice? Which are larger, the left-hand sums, right-hand sums, or midpoint sums? Can you explain it?

Assignment Part 3: Write commands that compute the left-hand sums and right-hand sum and midpoints sums numerically with 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560 subintervals. You can use a loop. What do you notice? Trapezoid Rule Unlike our previous rules, the trapezoid rule does not use approzimation by rectangles. As the name suggests trapezoids are used. This amounts to averaging the left hand value and right hand value for each box. There is a Maple command called trapezoid that computes the sums. evalf(trapezoid(f(x),x=1..4,5));

Assignment Part 4: Write commands that calculate the left hand, right hand, midpoint and trapezoid rules with 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560 subintervals. You can use a loop. What do you notice? Which are larger, smaller? Which are overestimates and which are underestimates of the integral? Why? Assignment Part 5: Work all the commands introduced today for the integral 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 with n=1, 2, 4, 8, 16, 32, 64 subintervals. Order in increasing order the left-hand sums, right-hand sums, midpoint sums and trapezoid sums. Explain you answer. Graph your sums for n=1 and n= 2 to explain your answer.