Triangle-AOC-inscribed-in-the-region-cut-from-the-parabola-y-x-2-by-the-line-y-a-2-Find-the-limit-of-ratio-of-the-area-of-the-triangle-to-the-area-of-the-parabolic-region-as-a-approaches-zero-

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

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

Answered by imjagoll last updated on 27/Jun/21

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

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