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Area between x=g(y) and y Axis

Let S be the ...

Question

Let $S$ be the region bounded by the curves $y = x^{3}$ and $y^{2} = x$ . The curve $y = 2 | x |$ divides $S$ into two regions of areas $R_{1}$ and $R_{2}$ . If $max {R_{1}, R_{2}} = R_{2}$ , then $\frac{R_{2}}{R_{1}}$ is equal to

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Solution

Given curves: $C_{1} : y = x^{3}, C_{2} : y^{2} = x$
and $C_{3} : y = 2 | x |$
$\Rightarrow C_{1}$ and $C_{2}$ intersect at $(1, 1)$ and $(0, 0)$
$\Rightarrow C_{2}$ and $C_{3}$ intersect at $(\frac{1}{4}, \frac{1}{2})$ and $(0, 0)$

Figure:

Now, $R_{1} + R_{2} = 1 \int 0 (\sqrt{x} - x^{3}) d x = \frac{2}{3} - \frac{1}{4} = \frac{5}{12}$
and $R_{1} = 1 / 4 \int 0 (\sqrt{x} - 2 x) d x = \frac{1}{48}$

So, $\frac{R_{1} + R_{2}}{R_{1}} = \frac{5 / 12}{1 / 48}$ $\Rightarrow 1 + \frac{R_{2}}{R_{1}} = 20$
$∴ \frac{R_{2}}{R_{1}} = 19$

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