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JEE MAIN - Mathematics (2022 - 24th June Morning Shift - No. 21)

Let S be the region bounded by the curves y = x3 and y2 = x. The curve y = 2|x| divides S into two regions of areas R1, R2. If max {R1, R2} = R2, then $${{{R_2}} \over {{R_1}}}$$ is equal to ______________.
Answer
19

Explanation

JEE Main 2022 (Online) 24th June Morning Shift Mathematics - Area Under The Curves Question 70 English Explanation
$C_{1}: y=x^{3}$

$C_{2}: y^{2}=x$

and $C_{3}=y=2|x|$

$C_{1}$ and $C_{2}$ intersect at $(1,1)$

$C_{2}$ and $C_{3}$ intersect at $\left(\frac{1}{4}, \frac{1}{2}\right)$

Clearly $R_{1}=\int_{0}^{1 / 4}(\sqrt{x}-2 x) d x=\frac{2}{3}\left(\frac{1}{8}\right)-\frac{1}{16}=\frac{1}{48}$

and $R_{1}+R_{2}=\int_{0}^{1}\left(\sqrt{x}-x^{3}\right) d x=\frac{2}{3}-\frac{1}{4}=\frac{5}{12}$

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

$\Rightarrow \frac{R_{2}}{R_{1}}=19$

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