MathProblemsBank

4.7 Variational calculus

Problem: Let \( M=C[a ; b]- \) be the set of all continuous functions \( y(x) \), given on the interval \( [a ; b] \). Given functional \( I[y(x)]=\pi \int_{a}^{b} y^{2}(x) d x \). Select three different functions that belong to \( C[a ; b] \), and obtain the corresponding values of \( I[y(x)] \), by taking specific \( \mathrm{a} \) and \( \mathrm{b} \).