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Question Number 102606 by bemath last updated on 10/Jul/20

what is the volume of region bounded by y =x^2 −2x and y=x that is rotated about y=4 ?

$${what}\:{is}\:{the}\:{volume}\:{of}\:{region} \\ $$$${bounded}\:{by}\:{y}\:={x}^{\mathrm{2}} −\mathrm{2}{x}\:{and} \\ $$$${y}={x}\:{that}\:{is}\:{rotated}\:{about} \\ $$$${y}=\mathrm{4}\:? \\ $$

Commented by bemath last updated on 10/Jul/20

Answered by Ar Brandon last updated on 10/Jul/20

Volume , V=π∫_0 ^3 {(x^2 −2x−4)^2 −(x−4)^2 }dx

$$\mathrm{Volume}\:,\:\mathrm{V}=\pi\int_{\mathrm{0}} ^{\mathrm{3}} \left\{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2x}−\mathrm{4}\right)^{\mathrm{2}} −\left(\mathrm{x}−\mathrm{4}\right)^{\mathrm{2}} \right\}\mathrm{dx} \\ $$

Answered by bobhans last updated on 10/Jul/20

vol = π∫_0 ^3 (4^2 −(x^2 −2x)^2 )−(4^2 −x^2 ) dx = π∫_0 ^3 (x^2 −(x^2 −2x)^2 dx = π∫_0 ^3 (x^2 +x^2 −2x)(2x)dx =4π∫_0 ^3 x(x^2 −x)dx = 4π∫_0 ^3 (x^3 −x^2 ) dx = 4π {(1/4)x^4 −(1/3)x^3 }_0 ^3 =4π {((81)/4)−((27)/3)} =4π {((45)/4)} = 45π

$${vol}\:=\:\pi\overset{\mathrm{3}} {\int}_{\mathrm{0}} \left(\mathrm{4}^{\mathrm{2}} −\left({x}^{\mathrm{2}} −\mathrm{2}{x}\right)^{\mathrm{2}} \right)−\left(\mathrm{4}^{\mathrm{2}} −{x}^{\mathrm{2}} \right)\:{dx} \\ $$$$=\:\pi\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\left({x}^{\mathrm{2}} −\left({x}^{\mathrm{2}} −\mathrm{2}{x}\right)^{\mathrm{2}} \:{dx}\:\right. \\ $$$$=\:\pi\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\left({x}^{\mathrm{2}} +{x}^{\mathrm{2}} −\mathrm{2}{x}\right)\left(\mathrm{2}{x}\right){dx} \\ $$$$=\mathrm{4}\pi\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}{x}\left({x}^{\mathrm{2}} −{x}\right){dx}\:=\:\mathrm{4}\pi\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\:\left({x}^{\mathrm{3}} −{x}^{\mathrm{2}} \right)\:{dx} \\ $$$$=\:\mathrm{4}\pi\:\left\{\frac{\mathrm{1}}{\mathrm{4}}{x}^{\mathrm{4}} −\frac{\mathrm{1}}{\mathrm{3}}{x}^{\mathrm{3}} \right\}_{\mathrm{0}} ^{\mathrm{3}} =\mathrm{4}\pi\:\left\{\frac{\mathrm{81}}{\mathrm{4}}−\frac{\mathrm{27}}{\mathrm{3}}\right\}\: \\ $$$$=\mathrm{4}\pi\:\left\{\frac{\mathrm{45}}{\mathrm{4}}\right\}\:=\:\mathrm{45}\pi \\ $$$$ \\ $$

Commented by Ar Brandon last updated on 10/Jul/20

What difference will it make if it was rather rotated about y=0 with respect to your solution ? ��

Commented by bemath last updated on 10/Jul/20

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