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Question Number 144638 by liberty last updated on 27/Jun/21

$$\mathrm{Triangle}\mathrm{AOC}\mathrm{inscribed}$$$$\mathrm{in}\mathrm{the}\mathrm{region}\mathrm{cut}\mathrm{from}$$$$\mathrm{the}\mathrm{parabola}\mathrm{y}={\mathrm{x}}^2\mathrm{by}\mathrm{the}$$$$\mathrm{line}\mathrm{y}={\mathrm{a}}^2.\mathrm{Find}\mathrm{the}\mathrm{limit}$$$$\mathrm{of}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{area}\mathrm{of}\mathrm{the}$$$$\mathrm{triangle}\mathrm{to}\mathrm{the}\mathrm{area}\mathrm{of}\mathrm{the}$$$$\mathrm{parabolic}\mathrm{region}\mathrm{as}\mathrm{a}\mathrm{approaches}$$$$\mathrm{zero}$$

Answered by imjagoll last updated on 27/Jun/21

$$\mathrm{area}△\mathrm{AOC}=\frac{1}{2}.2\mathrm{a}.{\mathrm{a}}^2={\mathrm{a}}^3$$$$\mathrm{area}\mathrm{parabolic}=\frac{2}{3}.2\mathrm{a}.{\mathrm{a}}^2=\frac{4}{3}{\mathrm{a}}^3$$$$\underset{\mathrm{a}\to 0}{\lim }\frac{{\mathrm{a}}^3}{\left(\frac{4}{3}{\mathrm{a}}^3\right)}=\frac{3}{4}$$

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