 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/14/g/solution

In the convention with constant term

a_{0} /2

\int_{0}^{2 π} y (x)^{2} d x = π [\frac{a _{0}^{2}}{2} + \sum_{n = 1}^{\infty} (a_{n}^{2} + b_{n}^{2})] .

Only the first harmonic contributes here, so

\int_{0}^{2 π} y (x)^{2} d x = π [(- 3)^{2} + 1^{2}] = 4 π .

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