Menu Close

If-y-sin-x-x-0-to-x-2pi-is-revolved-about-the-x-axis-find-the-surface-of-the-solid-of-revolution-




Question Number 87298 by ajfour last updated on 03/Apr/20
If y=sin x ,  x=0 to x=2π is  revolved about the x-axis, find  the surface of the solid of  revolution.
$${If}\:{y}=\mathrm{sin}\:{x}\:,\:\:{x}=\mathrm{0}\:{to}\:{x}=\mathrm{2}\pi\:{is} \\ $$$${revolved}\:{about}\:{the}\:{x}-{axis},\:{find} \\ $$$${the}\:{surface}\:{of}\:{the}\:{solid}\:{of} \\ $$$${revolution}. \\ $$
Answered by ajfour last updated on 04/Apr/20
A=2∫_0 ^( π) (2πy)ds    ds=dx×sec θ     tan θ=(dy/dx) = cos x  A=4π∫_0 ^( π) (sin x)((√(1+cos^2 x)) dx)     let  cos x=t  ⇒  −sin xdx=dt   A=4π∫_(−1) ^(  1) (√(1+t^2 )) dt      =8π∫_0 ^( 1) (√(1+t^2 )) dt     = 8π{(t/2)(√(1+t^2 ))+(1/2)ln ∣t+(√(1+t^2 ))∣ }∣_0 ^1     = 8π[(1/( (√2)))+(1/2)ln (1+(√2))]    A=4𝛑[(√2)+ln (1+(√2))].
$${A}=\mathrm{2}\int_{\mathrm{0}} ^{\:\pi} \left(\mathrm{2}\pi{y}\right){ds} \\ $$$$\:\:{ds}={dx}×\mathrm{sec}\:\theta \\ $$$$\:\:\:\mathrm{tan}\:\theta=\frac{{dy}}{{dx}}\:=\:\mathrm{cos}\:{x} \\ $$$${A}=\mathrm{4}\pi\int_{\mathrm{0}} ^{\:\pi} \left(\mathrm{sin}\:{x}\right)\left(\sqrt{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} {x}}\:{dx}\right) \\ $$$$\:\:\:{let}\:\:\mathrm{cos}\:{x}={t}\:\:\Rightarrow\:\:−\mathrm{sin}\:{xdx}={dt} \\ $$$$\:{A}=\mathrm{4}\pi\int_{−\mathrm{1}} ^{\:\:\mathrm{1}} \sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$$\:\:\:\:=\mathrm{8}\pi\int_{\mathrm{0}} ^{\:\mathrm{1}} \sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$$\:\:\:=\:\mathrm{8}\pi\left\{\frac{{t}}{\mathrm{2}}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\mid{t}+\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\mid\:\right\}\mid_{\mathrm{0}} ^{\mathrm{1}} \\ $$$$\:\:=\:\mathrm{8}\pi\left[\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)\right] \\ $$$$\:\:\boldsymbol{{A}}=\mathrm{4}\boldsymbol{\pi}\left[\sqrt{\mathrm{2}}+\mathrm{ln}\:\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)\right]. \\ $$$$ \\ $$
Commented by mr W last updated on 04/Apr/20
you are right sir! thanks!
$${you}\:{are}\:{right}\:{sir}!\:{thanks}! \\ $$
Commented by ajfour last updated on 04/Apr/20
You might like Q.87296 Sir.
$${You}\:{might}\:{like}\:{Q}.\mathrm{87296}\:{Sir}. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *