The main idea of this section is numerical approximations of integrals using computers. The first technique presented is Euler's method. This method entails calculation x and the velocity of x multiple times over very small steps. This is the simplest form of numerical integration, and is the least accurate. The methods increase in order up to the fourth-order Runge-Kutta method which is a good balance of accuracy and non-tediousness. This method involves finding 4 values and then x(n+1) is a number based on these values. Avoiding excessively small step-sizes is important because computers round-off at every calculation.
Challenges
My main question from this section is: how do you get a computer to do this for you? The book mentions writing your own numerical integration routine but I'm sure there is a program already available. Are we using numerical integration to check answers, or to find solutions when other methods don't work?
Reflections
I like the idea of numerical integration because it brings back fond memories of checking my answers in Calculus using the fnInt function on my calculator. I have a definite interest in computer science, also, so I am looking forward to class tomorrow and to working on this section.
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