1. Aims: To be able to use the Mid-ordinate rule to calculate an estimate for the area. To be able to check your estimated answer with the acurate one using your calculator. Numerical Methods Lesson 3

2. Intro The Mid-ordinate Rule y x The Mid-ordinate Rule is similar to the Trapezium Rule. It uses a series of rectangles of equal width to estimate the area under a graph between two points a and b. The height of each rectangle is determined by the height of the curve at the m_________________ of the interval.  hy 1 + hy 2 + hy 3 +... + hy n-1 + hy n a b Area  h(y 1 + y 2 + y 3 +... + y n-1 + y n ) h h h h O ynyn y1y1 y2y2 y3y3

3. The Mid-ordinate Rule Area  h(y 1 + y 2 + y 3 +... + y n-1 + y n ) x5.56.57.5 y = f(x) 12.25 30.25 42.25 56.25 20.25 Approximate using the mid-ordinate rule with 5 strips. Example Question 1 We tabulate as below. x y = f(x) 2 4 6 8 01 357 A  1(12.25 + 20.25 + 30.25 + 42.25 + 56.25) = 161.25 3.54.5

4. The Mid-ordinate Rule Area  h(y 1 + y 2 + y 3 +... + y n-1 + y n ) x y = f(x) x1.752.252.75 y =f(x) 0.9412 0.3902 0.2462 0.1649 0.1168 Approximate using the mid-ordinate rule with 6 strips. Example Question 2 We tabulate as below. A  ½(0.9412 + 0.64 + 0.3902 + 0.2462 + 0.1649 + 0.1168) = 1.25 (2 dp) 0.64 0.5 1 1.5 2 2.5 3 0.250.751.25

5. The Mid-ordinate Rule Area  h(y 1 + y 2 + y 3 +... + y n-1 + y n ) x y = f(x) 3.375 42.875 91.125 Approximate using the mid-ordinate rule with 4 strips. Example Question 3 15.625 We tabulate as below. A  1(3.375 + 15.625 + 42.875 + 91.125) = 153 1.52.53.54.5 Check ans approximately correct with your calculator: Run Mode OPTN, F4 (calc), F4 (intergrate) enter X^3,1,5) press EXE 156

6. The Mid-ordinate Rule Area  h(y 1 + y 2 + y 3 +... + y n-1 + y n ) x y = f(x) 0.25 Example Question 4 1 A   /3(0.25 + 1 + 0.25) = 1.57 (2 dp) Approximate using the mid-ordinate rule with 3 strips. We tabulate as below.  /3 0 2  /3  5  /6  /6  /2

7. The Mid-ordinate Rule Question 1: Approximate using the mid-ordinate rule with 4 strips.  21 Question 2: Approximate using the mid-ordinate rule with 5 strips.  0.6919 Question 3: Approximate using the mid-ordinate rule with 4 strips.  1.736 Question 4: Approximate using the mid-ordinate rule with 4 strips.  1.40 Homework, do exercise C page 139

8. Exercises using the mid-ordinate rule with 4 strips, giving your answer to 3 d.p. How can your answer be improved? 1. Estimate rule with 3 strips. Give your answer to 3 s.f. 2. Estimateusing the mid-ordinate N.B. Radians !

9. Solutions The answer can be improved by using more strips. 1.

10. Solutions

11. The red shaded areas should be included but are not. The blue shaded areas are not under the curve but are included in the rectangle. The following sketches show sample rectangles where the mid-ordinate rule under- and over-estimates the area. Under-estimates ( concave upwards ) Over-estimates ( concave downwards )

16. Past Exam Question

18. Questions about the integral ∫ 0 2 √(1+x 3 )dx. The value of this integral, correct to four decimal places, is 3.2413. The percentage error in the use of the mid-ordinate rule in question 5 is (a) -0.65% (b) -0.66% (c) 1.30% (d) 1.28% Integration on GDC

19. Multi guess worksheet

20. Three minutes to answer 3 out of these 10 questions on your white boards. Discuss in pairs your comments – 2 minutes Share with class What I liked most about this lesson was… I was surprised by… After this lesson I feel… I might have learned more if… Today I learned… One thing I’m not sure about is… I was interested in… The most useful thing was… One thing I want to find out more about is… What still puzzles me is… Here’s a ‘fun’ picture of the mid- ordinate rule

21. Worksheet Question 1: Approximate using the mid-ordinate rule with 4 strips. Question 2: Approximate using the mid-ordinate rule with 5 strips. Question 4: Approximate using the mid-ordinate rule with 4 strips. Question 3: Approximate using the mid-ordinate rule with 4 strips. Question 1: Approximate using the mid-ordinate rule with 4 strips. Question 2: Approximate using the mid-ordinate rule with 5 strips. Question 4: Approximate using the mid-ordinate rule with 4 strips. Question 3: Approximate using the mid-ordinate rule with 4 strips.