December 2021 C 3 Simpsons Rule of Approximate

December 2021 C 3 Simpson’s Rule of Approximate Integration Objective: To be able to use Simpson’s Rule to find approximations of integrals.

Simpson’s Rule of Approximate Integration We are going to use a series of parabolas in order to determine the area under a curve. Firstly we need a general formula for the area under a parabola: Take any three points, evenly spaced. c + x (-h, y 0) x a y= Draw a parabola through them. +b (h, y 2) General equation of parabola is: y = ax 2 + bx + c Integrate to find the area beneath the parabola: 2 (0, y 1 ) Call the distance between the points h and we can determine the y co-ordinates of each point. y 0 = ah 2 – bh + c y 1 = c y 2 = ah 2 + bh + c Notice that y 0 + 4 y 1 + y 2 = 2 ah 2 +6 c To be able to use Simpson’s Rule to find approximations of integrals.

Simpson’s Rule of Approximate Integration Now we can use this result to find the area under real curves: Approximate the area under the curve shown between the limits a and b by using three points. y 0 Now add two more points. a y 1 y y 21 y 3 y 2 y 4 b Points can be added in pairs indefinitely. . . This rule is known as Simpson’s Rule, and gives us an approximation for the integral of a function. Note that there are n strips, and therefore (n + 1) ordinates. To be able to use Simpson’s Rule to find approximations of integrals.

Simpson’s Rule of Approximate Integration Example Use Simpson’s rule with 5 ordinates to find the approximate value of your answer to three decimal places. giving To be able to use Simpson’s Rule to find approximations of integrals.

Simpson’s Rule of Approximate Integration Example Use Simpson’s rule with six equal strips to find an approximate value for giving your answer correct to 4 decimal places. To be able to use Simpson’s Rule to find approximations of integrals.

Simpson’s Rule of Approximate Integration Calculate approximations of the following integrals: 1. using 5 ordinates 2. using 4 strips 3. using 7 ordinates To be able to use Simpson’s Rule to find approximations of integrals.