Riley hobson bence · Problem 2.47

Preliminary calculus

Mathematics — Methods · integral-inequalities

Problem

By noting that for 0≤η≤10 \leq \eta \leq 1, η1/2≥η3/4≥η\eta^{1/2} \geq \eta^{3/4} \geq \eta, prove that 23≤1a5/2∫0a(a2−x2)3/4 dx≤π4.\frac{2}{3} \leq \frac{1}{a^{5/2}} \int_0^a (a^2 - x^2)^{3/4}\, dx \leq \frac{\pi}{4}.

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