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Question Number 139272 by bobhans last updated on 25/Apr/21

$$$$ What is the area bounded by the parabola y^2=4ax and x^2 = 4ay?\n

Answered by EDWIN88 last updated on 25/Apr/21

$$\mathrm{case}\left(1\right)\mathrm{If}\mathrm{a}>0$$ $$\Rightarrow \mathrm{Area}=\underset{0}{\stackrel{4\mathrm{a}}{\int }}(2\sqrt{\mathrm{ax}}-\frac{{\mathrm{x}}^2}{4\mathrm{a}})\mathrm{dx}$$ $$\Rightarrow \mathrm{Area}={[\frac{4}{3}\sqrt{\mathrm{a}}{\mathrm{x}}^{3/2}-\frac{{\mathrm{x}}^3}{12\mathrm{a}}]}_0^{4\mathrm{a}}$$ $$\Rightarrow \mathrm{Area}=\frac{4}{3}.4\mathrm{a}\sqrt{{4\mathrm{a}}^2}-\frac{{64\mathrm{a}}^3}{12\mathrm{a}}$$ $$\Rightarrow \mathrm{Area}=\frac{{32\mathrm{a}}^2}{3}-\frac{{16\mathrm{a}}^2}{3}=\frac{{16\mathrm{a}}^2}{3}.$$

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