# **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 --- # **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 $h = \frac{b-a}{n}$ --- # **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.