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Question Number 184287 by Tawa11 last updated on 04/Jan/23

If the area enclosed between the curves y=x² and the line y = 2x is rotated round the x-axis through 4 right angles, find the volume of the solid generated

Answered by mr W last updated on 04/Jan/23

$$y_1=2x$$$$y_2=x^2$$$$2x=x^2\Rightarrow x=0,x=2$$$$V={\int }_0^2\pi (y_1^2-y_2^2)dx$$$$={\int }_0^2\pi (4x^2-x^4)dx$$$$=\pi {[\frac{4x^3}{3}-\frac{x^5}{5}]}_0^2$$$$=\pi (\frac{4\times 8}{3}-\frac{32}{5})$$$$=\frac{64\pi }{15}$$

Commented by Tawa11 last updated on 04/Jan/23

$$\mathrm{I}\mathrm{appreciate}\mathrm{sir}.$$

Answered by mr W last updated on 04/Jan/23

$$V_1=2\pi A_1{y}_{S1}=2\pi \times \frac{2\times 4}{2}\times \frac{4}{3}=\frac{32\pi }{3}$$$$V_2=2\pi A_2{y}_{S2}=2\pi \times \frac{2\times 4}{3}\times \frac{3\times 4}{10}=\frac{32\pi }{5}$$$$V=\frac{32\pi }{3}-\frac{32\pi }{5}=\frac{64\pi }{15}$$

Commented by Tawa11 last updated on 05/Jan/23

$$\mathrm{Thanks}\mathrm{for}\mathrm{your}\mathrm{time}\mathrm{sir}.$$$$\mathrm{God}\mathrm{bless}\mathrm{you}.$$

Commented by Tawa11 last updated on 05/Jan/23

$$\mathrm{Can}\mathrm{I}\mathrm{see}\mathrm{the}\mathrm{diagram}\mathrm{sir}?$$

Commented by mr W last updated on 05/Jan/23

$$diagramforwhat?$$

Commented by Tawa11 last updated on 05/Jan/23

$$\mathrm{Graph}\mathrm{I}\mathrm{mean}\mathrm{sir}.$$$$\mathrm{For}\mathrm{the}\mathrm{volume}\mathrm{as}\mathrm{the}\mathrm{area}\mathrm{rotate}.$$$$\mathrm{Is}\mathrm{ut}\mathrm{possible}\mathrm{sir}.$$

Commented by mr W last updated on 05/Jan/23

$$iassumedyouknowhowthegraph$$$$fory=2xandy=x^{2}is.$$

Commented by Tawa11 last updated on 05/Jan/23

$$\mathrm{Yes}\mathrm{sir}.\mathrm{Very}\mathrm{well}.$$

Commented by mr W last updated on 05/Jan/23

$$seeQ158209whichyouhaveread.$$

Commented by Tawa11 last updated on 05/Jan/23

$$\mathrm{Seen}\mathrm{sir},\mathrm{I}\mathrm{understand}\mathrm{now}.$$$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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