Calculate the area of the circle by the indicated method.The...

Calculate the area of the circle by the indicated method.The lengths of parallel chords of a circle that are 0.250 in. apart are given in the following table. The diameter of the circle is 2.000 in. The distance shown is the distance from one end of a diameter. (TABLE CAN'T COPY) Using the formula $A=\pi r^{2},$ the area of the circle is 3.14 in.². Find the area of the circle using Simpson's rule and the same table values as in Exercise $19 .$ Explain why the value found is closer to 3.14 in. $^{2}$ than the value found in Exercise $19 .$

Calculate the area of the circle by the indicated method.The lengths of parallel chords of a circle that are 0.250 in. apart are given in the following table. The diameter of the circle is 2.000 in. The distance shown is the distance from one end of a diameter. (TABLE CAN'T COPY) Using the formula $A=\pi r^{2},$ the area of the circle is 3.14 in.².
Find the area of the circle using Simpson's rule and the same table values as in Exercise $19 .$ Explain why the value found is closer to 3.14 in. $^{2}$ than the value found in Exercise $19 .$

Key Concepts

Numerical Integration

The process of approximating the value of a definite integral by using discrete data points rather than an analytic expression. It is essential when dealing with empirical data or functions that are difficult to integrate analytically. Techniques like Simpson's rule help in obtaining a close approximation by systematically reducing the error inherent in replacing a continuous process with discrete summation.

Error Analysis in Numerical Methods

Refers to assessing and understanding the difference between the approximate numerical solution and the true value. Simpson's rule tends to give more accurate results compared to other numerical methods because its error term includes a higher order derivative of the function being integrated. This means that, as long as the function is smooth, the approximation error decreases significantly, making it a preferable method in calculating areas like that of a circle.

Simpson's Rule

A numerical integration method that approximates the integral by fitting parabolic arcs through sets of three consecutive data points. This rule improves the accuracy over simpler methods like the trapezoidal rule by effectively capturing the curvature of the function. It is especially efficient when the integrand is approximately quadratic over the subintervals, making it useful in estimating areas under curves when data is sampled at equally spaced intervals.

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