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limit of sum of integrals

Source: Romanian District Olympiad 2015, Grade XII, Problem 2

September 26, 2018
integralsreal analysiscalculuscontestsintegration

Problem Statement

a) Calculate ∫01xsin⁡(πx2)dx. \int_{0}^1 x\sin\left( \pi x^2\right) dx.
b) Calculate lim⁡n→∞1n∑k=0n−1k∫knk+1nsin⁡(πx2)dx. \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx.
Florin Stănescu
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