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Understand integrals as limits of sums and their connection to derivatives through the Fundamental Theorem of Calculus. This deck introduces antiderivatives, area under curves, and practical calculus ...
Numerical integration is used to approximate the value of definite integrals when an analytical solution is difficult or impossible to obtain. It involves dividing the area under a curve into smaller segments and summing their areas to estimate the total.
William Kahan is a mathematician known for his work in numerical analysis, particularly in the development of algorithms for numerical integration and error analysis. He introduced formulas that utilize sample points to improve the accuracy of numerical integration.
Kahan's numerical integration formulas use sample points such as 1, 3, 7, 15, etc. These points are chosen to optimize the accuracy of the integration by leveraging previous calculations.
The trapezoidal rule approximates the area under a curve by dividing the interval into smaller segments and approximating the area of each segment as a trapezoid. The total area is then the sum of the areas of these trapezoids.
The error associated with the trapezoidal rule is proportional to the square of the width of the subintervals used in the approximation. It is generally less accurate for functions that are not linear over the interval.
Simpson's rule improves upon the trapezoidal rule by using parabolic segments instead of linear segments to approximate the area under a curve. This results in a more accurate estimate, particularly for functions that are smooth and continuous.
The midpoint rule estimates the area under a curve by using the value of the function at the midpoint of each subinterval. This method can provide better accuracy than the trapezoidal rule for certain types of functions.
Considering the error in numerical integration methods is crucial because it helps determine the reliability and accuracy of the results. Understanding the error allows for adjustments in the method or the number of sample points to achieve desired precision.
Generally, increasing the number of sample points in numerical integration leads to greater accuracy, as it allows for a finer approximation of the area under the curve. However, this also increases computational cost.
The choice of function significantly affects the numerical integration process because functions with high variability or discontinuities can lead to larger errors in approximation. Smooth and continuous functions yield better results with numerical methods.
The total error in numerical integration for the function y = cos(x) over the interval [0, 1] is small because this interval is a complete period of the cosine function, allowing for accurate approximations.
The area under the curve y = f(x) from 0 to x is represented as f(x + Δx) = f(x) + Δf, where Δf is the additional area added as x increases. This relationship helps in understanding the incremental area as x changes.
Rectangles can be used to estimate the area under a curve by creating a series of rectangles that either enclose or approximate the area. The heights of the rectangles can be determined using left endpoints, right endpoints, or midpoints.
Odd and even functions have specific properties that can simplify the numerical integration process. For example, the integral of an odd function over a symmetric interval is zero, while even functions can be integrated more easily due to their symmetry.
The expected value is a measure of the central tendency of a random variable, calculated as the weighted average of all possible values, where the weights are the probabilities of each outcome. In numerical integration, it can be estimated using integration techniques.
The area under the curve representing a company's expense rate over time can be interpreted as the total expenses incurred during that period. This relationship helps in financial analysis and budgeting.
Using different intervals in numerical integration can affect the accuracy of the approximation. Smaller intervals generally yield more accurate results, but require more computational effort, while larger intervals may lead to greater errors.
Numerical integration can be applied in various real-world scenarios, such as calculating areas under curves in physics, estimating total revenue in economics, and analyzing data trends in statistics.
The limitations of numerical integration methods include potential inaccuracies due to function behavior, computational complexity, and the need for careful selection of sample points to minimize error.