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Today's calculation of Integral 150

Source: Kanazawa University entrance exam 1987

September 2, 2006
calculusintegrationlimitlogarithmscalculus computations

Problem Statement

Find the value of aa such that lim⁡n→∞32∫−a3a3(1−t3n)nt2 dt=2  (n=1,2,⋯ ).\lim_{n\to\infty}\frac{3}{2}\int_{-\sqrt[3]{a}}^{\sqrt[3]{a}}\left(1-\frac{t^{3}}{n}\right)^{n}t^{2}\ dt=\sqrt{2}\ \ (n=1,2,\cdots).
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