proj-calc-part2 - Numerical Calculus Project Part II:...

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Numerical Calculus Project Part II: Integration Context: Your next project will be all about numerical calculus. This project will be split into three parts. The first two will be preliminary tasks where you will become familiar with numerical differentiation and integration and write and test drive methods to compute numerical derivatives and integrals. The final part of the project will use the functions from the first two parts and bring them together with an interface and analysis. This is the second part. This part of the project requires you to use skills up to Exam 2 and what you have learned about data files. Theory Recall from calculus the concept of the integral. In this project, we'll be using the computer to help us compute definite integrals of functions, ultimately with the goal of finding the area under a curve between given bounds. As you hopefully learned in calculus, we can investigate mathematics via the rule of three -- analytically, graphically, and numerically. There are many cases where computing integrals analytically is very challenging, tedious, or impossible, so we'll once again use numerical techniques to investigate them. We call this numerical integration process quadrature . Ultimately, the same idea is at the heart of all of our quadrature methods: take the area under the curve and divide and conquer, i.e. break it up into smaller areas we can compute or approximate more easily and add them up to get an approximation of the area we want. You likely saw some of this material in your Calculus I experience. We'll begin there, with Riemann sums. Let's suppose we're aiming to find the definite integral of f ( x ) between the bounds of x = a and x = b . There are two basic starting strategies: Left Sum: Divide the interval [ a , b ] up into n parts, effectively dividing the area under f ( x ) into n parts as well. Use rectangles of width ( b-a )/ n to approximate the area. The height of each rectangle should be f ( x L ), where x L is the left endpoint of the interval. Right Sum: Similarly, divide the interval [ a , b ] up into n parts, effectively dividing the area under f ( x ) into n parts as well. Use rectangles of width ( b-a )/ n to approximate the area. This time, however, use f ( x R ), where x R is the right endpoint of the interval, as the height of each rectangle. Of course, a few rectangles rarely approximate areas under curved functions perfectly and these integrals are
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proj-calc-part2 - Numerical Calculus Project Part II:...

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