Question Number 123163 by aurpeyz last updated on 23/Nov/20
$${the}\:{curve}\:{y}={x}^{\mathrm{2}} +\mathrm{4}\:{is}\:{rotated}\:{one}\: \\ $$$${revolution}\:{about}\:{the}\:{x}−{axis}\:{between} \\ $$$${the}\:{limits}\:{x}=\mathrm{1}\:{and}\:{x}=\mathrm{4}.\:{determine} \\ $$$${the}\:{volume}\:{of}\:{the}\:{revolution}\:{produced} \\ $$
Answered by ajfour last updated on 23/Nov/20
$${V}=\int_{\mathrm{1}} ^{\:\:\mathrm{4}} \pi{y}^{\mathrm{2}} {dx}\:=\pi\int_{\mathrm{1}} ^{\:\mathrm{4}} \left({x}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{2}} {dx} \\ $$$$\:=\:\pi\left(\frac{{x}^{\mathrm{5}} }{\mathrm{5}}+\frac{\mathrm{8}{x}^{\mathrm{3}} }{\mathrm{3}}+\mathrm{16}{x}\right)\mid_{\mathrm{1}} ^{\mathrm{4}} \\ $$$$\:\:{Volume}\:=\:\frac{\mathrm{2103}\pi}{\mathrm{5}} \\ $$