Menu Close

If-part-of-the-curve-y-x-1-from-x-1-and-x-2-is-rotated-completely-about-the-y-axis-find-the-volume-of-the-solid-form-Mastermind-




Question Number 170412 by Mastermind last updated on 23/May/22
If part of the curve y=x+1 from x=1  and x=2 is rotated completely about  the y−axis. find the volume of the  solid form.    Mastermind
$${If}\:{part}\:{of}\:{the}\:{curve}\:{y}={x}+\mathrm{1}\:{from}\:{x}=\mathrm{1} \\ $$$${and}\:{x}=\mathrm{2}\:{is}\:{rotated}\:{completely}\:{about} \\ $$$${the}\:{y}−{axis}.\:{find}\:{the}\:{volume}\:{of}\:{the} \\ $$$${solid}\:{form}. \\ $$$$ \\ $$$${Mastermind} \\ $$
Answered by aleks041103 last updated on 23/May/22
The resulted shape is a trincated cone.  r_1 =1, r_2 =2  h=∣y_2 −y_1 ∣=1  ⇒V=(1/3)πh(r_1 ^2 +r_2 ^2 +r_1 r_2 )=  =(1/3)π(1+4+2)=((7π)/3)  ⇒V=((7π)/3)
$${The}\:{resulted}\:{shape}\:{is}\:{a}\:{trincated}\:{cone}. \\ $$$${r}_{\mathrm{1}} =\mathrm{1},\:{r}_{\mathrm{2}} =\mathrm{2} \\ $$$${h}=\mid{y}_{\mathrm{2}} −{y}_{\mathrm{1}} \mid=\mathrm{1} \\ $$$$\Rightarrow{V}=\frac{\mathrm{1}}{\mathrm{3}}\pi{h}\left({r}_{\mathrm{1}} ^{\mathrm{2}} +{r}_{\mathrm{2}} ^{\mathrm{2}} +{r}_{\mathrm{1}} {r}_{\mathrm{2}} \right)= \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}\pi\left(\mathrm{1}+\mathrm{4}+\mathrm{2}\right)=\frac{\mathrm{7}\pi}{\mathrm{3}} \\ $$$$\Rightarrow{V}=\frac{\mathrm{7}\pi}{\mathrm{3}} \\ $$
Commented by Mastermind last updated on 24/May/22
  Thanks
$$ \\ $$$${Thanks} \\ $$
Answered by MikeH last updated on 23/May/22
V = π∫_a ^b y^2  dx  ⇒ V = π∫_1 ^2 (x^2 +2x+1)dx  V = π[(x^3 /3) + x^2 +x]_1 ^2   V = ((19π)/3) cubic units
$${V}\:=\:\pi\int_{{a}} ^{{b}} {y}^{\mathrm{2}} \:{dx} \\ $$$$\Rightarrow\:{V}\:=\:\pi\int_{\mathrm{1}} ^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}\right){dx} \\ $$$${V}\:=\:\pi\left[\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\:+\:{x}^{\mathrm{2}} +{x}\right]_{\mathrm{1}} ^{\mathrm{2}} \\ $$$${V}\:=\:\frac{\mathrm{19}\pi}{\mathrm{3}}\:\mathrm{cubic}\:\mathrm{units} \\ $$
Commented by Mastermind last updated on 24/May/22
  Thank you
$$ \\ $$$${Thank}\:{you}\: \\ $$$$ \\ $$

Leave a Reply

Your email address will not be published.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