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 all values in the table. Explain why the value found is closer to 3.14 in. ' than the value found in Exercise 21

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 all values in the table. Explain why the value found is closer to 3.14 in. ' than the value found in Exercise 21

Key Concepts

Simpson's Rule

Simpson's Rule is a numerical method for approximating the integral of a function by replacing segments of the curve with parabolic arcs. It uses quadratic polynomials to approximate the function over small subintervals, providing an effective and accurate estimate when the function is smooth. This method requires an even number of subintervals and applies specific weights to the function values at the endpoints and midpoints of these segments.

Numerical Integration

Numerical integration involves methods for estimating the value of a definite integral when an antiderivative is difficult to obtain or experimental data is used. Techniques like Simpson's Rule are employed to handle discrete data points, transforming them into a continuous approximation of the function. This allows for the calculation of areas under curves when only tabulated values are available.

Error Analysis in Numerical Methods

Error analysis in numerical integration focuses on understanding and minimizing the difference between the numerical approximation and the exact integral. Simpson's Rule, by utilizing quadratic approximations, typically yields a lower error compared to simpler methods like the Trapezoidal Rule, especially for functions that exhibit smooth, curved behavior. This is why the value calculated using Simpson's Rule can be closer to the true area, such as the expected area of a circle.

Area Calculation of a Circle

The area of a circle is traditionally calculated using the formula A = ?r². However, in practical applications where measurements are taken at discrete points (such as chord lengths across a circle), numerical integration can be used to approximate the area. By integrating these measurements along the diameter of the circle, the method can closely replicate the actual area when the integration method effectively captures the curvature of the circle.

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