The goal is to implement and compare the basic Quadrature Rules for I = _a^b F(x) dx , the integral of a given function F(x) over an interval [a,b], using M subintervals.

The integrand F(x) should be coded as a Matlab function FCN in a separate file FCN.m 1. Implement the Rectangle Rule as a Matlab function in a code RectRule.m , with both Left and with Right Heights. a. Inputs to the function RectRule should be a, b, M b. The code should print out: M and value of RL = Rectangle Rule approximation using Left heights and value of RR = Rectangle Rule approximation using Right heights, nicely formated, like: RectRule for M= VALUE : RL= VALUE , RR= VALUE c. Debug on ₀¹ 3 dx with M=4. d. Debug on ₀¹ (2x1) dx with M=4. e. Test it on ₀¹ 3(1x²)dx with M=1000. Looks like rather large M is needed to get a decent approximation... 2. Implement the Midpoint Rule as a Matlab function in a code MidptRule.m Repeat a - e above to debug and test the Midpoint Rule. 3. Implement the Trapezoidal Rule as a Matlab function in a code TrapzRule.m Repeat a - e above to debug and test the Trapezoidal Rule. 4. Create a Matlab function Quadr.m with input arguments a, b, M , which A. calls each of your quadrature rules (with the same a, b, M), B. computes the error of each rule by comparing with I_exact (see below) C. prints a (nicely formatted) table like: Comparison of Quadrature Rules using M= VALUE subintervals: Rule Value Error 5. Value of Pi: How can we produce (to machine accuracy) if we need it in a computation, in any language ? Very easily, from the arctan(x) function: since tan(/4) = 1, we have: PI = 4*atan(1.e0) In fact, this is the best way to set the value of PI = in a code, in any language.

Insert the formula as I_exact in your "Quadr.m" so you can compute and print the errors. 6. Use your Quadr.m to find an approximation to the number PI from the formula: PI = 4 ₀¹ (1x²) dx a. Using M = 1000, how many correct digits does TrapzRule produce ? b. Using M = 100000, how many correct digits does TrapzRule produce ? c. What is the connection between the formulas for PI in 5. and in 6. ? 7. What to submit: In a single plain text file "Lab9.m" insert the following: 0. % NAME , Date , Lab9 1. Answers to questions 6a, 6b, 6c (as comments) If your results are unexpected, any guesses as to what may be happening ? % =============================================== (separator) 2. your "FCN.m" file, showing coding of the integrand for PI in 6. % =============================================== (separator) 3. your "Quadr.m" file % =============================================== (separator) 4. your "TrapzRule.m" file % =============================================== (separator)