Question Bank • QB_EM_128 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus

Subtopic: Fourier, Taylor and MacLaurin series

Question Statement:

In the Fourier series expansion of the function $f(x) = x^2$ on the symmetric interval $[-\pi, \pi]$, calculate the constant term $a_0 = \frac{1}{\pi} \int_{-\pi}^\pi x^2 dx$. (Express as a multiple of $\pi^2$, i.e. $a_0 / \pi^2$).

Answer Key:

0.667

Step-by-Step Solution & Explanation

$a_0 = \frac{1}{\pi} \int_{-\pi}^\pi x^2 dx = \frac{2}{\pi} \int_0^\pi x^2 dx = \frac{2}{\pi} \left[ \frac{x^3}{3} \right]_0^\pi = \frac{2}{\pi} \frac{\pi^3}{3} = \frac{2}{3} \pi^2$ So $a_0 / \pi^2 = \frac{2}{3} \approx 0.667$.
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