Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on the following concepts:

• Numerical integration methods

• Trapezoidal rule for approximating definite integrals

• Simpson's $\frac{1}{3}$ rule for approximating definite integrals

• Step size calculation and interval subdivision

• Definite integrals of algebraic, rational, and trigonometric functions
• Composite numerical integration techniques

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Important Formulas

Below are the key formulas used in this exercise:

Trapezoidal Rule: ${\int }_a^{b}f(x)dx\approx \frac{h}{2}\left[(y_0+y_n)+2{\sum }_{i=1}^{n-1}{y}_i\right]$

where $h=\frac{b-a}{n}$ is the step size, $n$ is the number of subintervals, and $y_i=f(x_i)$

Simpson's $\frac{1}{3}$ Rule: ${\int }_a^{b}f(x)dx\approx \frac{h}{3}\left[(y_0+y_n)+4(y_1+y_3+\cdots +y_{n-1})+2(y_2+y_4+\cdots +y_{n-2})\right]$

where $n$ is even and $h=\frac{b-a}{n}$

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Summary

This exercise covers numerical approximation of definite integrals using the Trapezoidal and Simpson's $\frac{1}{3}$ rules. The Trapezoidal rule approximates the region under curves using trapezoids (linear approximations), while Simpson's rule uses parabolic arcs for higher accuracy.

Key strategies include: determining the correct step size $h=\frac{b-a}{n}$, evaluating functions at equally spaced nodes, and applying proper weighting coefficients (1-2-2-...-2-1 for Trapezoidal; 1-4-2-4-...-2-4-1 for Simpson's). The exercise demonstrates these methods on various integrands including polynomials, rational functions, and trigonometric functions, highlighting the trade-off between computational complexity and approximation accuracy.