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598

Part of 2010 Today's Calculation Of Integral

Problems(1)

Today's calculation of Integral 598

Source:

4/27/2010
For a constant aa, denote C(a)C(a) the part x≥1x\geq 1 of the curve y=x2−1+axy=\sqrt{x^2-1}+\frac{a}{x}.
(1) Find the maximum value a0a_0 of aa such that C(a)C(a) is contained to lower part of y=xy=x, or y<xy<x.
(2) For 0<θ<π20<\theta <\frac{\pi}{2}, find the volume V(θ)V(\theta) of the solid VV obtained by revoloving the figure bounded by C(a0)C(a_0) and three lines y=x, x=1, x=1cos⁡θy=x,\ x=1,\ x=\frac{1}{\cos \theta} about the xx-axis.
(3) Find lim⁡θ→π2−0V(θ)\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta).
1992 Tokyo University entrance exam/Science, 2nd exam
calculusintegrationtrigonometrylimitcalculus computations