Question Number 25745 by rita1608 last updated on 13/Dec/17
$${find}\:{the}\:{volume}\:{of}\:{the}\:{solid}\: \\ $$$${generated}\:{by}\:{thr}\:{revolution}\:{of}\:{the} \\ $$$${curve}\:{y}\left({x}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)={a}^{\mathrm{3}} \:{about}\:{itd}\: \\ $$$${asymptote}. \\ $$
Answered by jota@ last updated on 15/Dec/17
$${asintota}\:{y}=\mathrm{0}\: \\ $$$${V}=\mathrm{2}\int_{\mathrm{0}} ^{\infty} \pi{y}^{\mathrm{2}} {dx} \\ $$$${V}=\mathrm{2}\pi{a}^{\mathrm{6}} \int_{\mathrm{0}} ^{\infty} \frac{{dx}}{\left({x}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)^{\mathrm{2}} }. \\ $$$$\:{x}={atanz}\:\:\:{dx}={asec}^{\mathrm{2}} {zdz} \\ $$$$\:{x}^{\mathrm{2}} +{a}^{\mathrm{2}} ={a}^{\mathrm{2}} \left({tan}^{\mathrm{2}} {z}+\mathrm{1}\right)={a}^{\mathrm{2}} {sec}^{\mathrm{2}} {z} \\ $$$${V}=\mathrm{2}\pi{a}^{\mathrm{6}} \int_{\mathrm{0}} ^{\boldsymbol{\pi}/\mathrm{2}} \frac{{cos}^{\mathrm{2}} {zdz}}{{a}^{\mathrm{3}} } \\ $$$$\:\:=\mathrm{2}\pi{a}^{\mathrm{3}} \int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{\mathrm{1}+{cos}\mathrm{2}{z}}{\mathrm{2}}{dz} \\ $$$$\:\:=\pi{a}^{\mathrm{3}} \left({z}+\frac{{sin}\mathrm{2}{z}}{\mathrm{2}}\right)\mid_{\mathrm{0}} ^{\pi/\mathrm{2}} \\ $$$$\:\:=\frac{\pi^{\mathrm{2}} {a}^{\mathrm{3}} }{\mathrm{2}}. \\ $$$$ \\ $$