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 and R2. If max {R1,R2} = R2 , then R2R1is equal to ____ .

S=∫01 x−x3=2x3/23−x4410R1+R2=512R1=∫01/4 (x−2x)dx=2x3/23−x201/4=148∴R2=512-148=1948So, R2R1=19

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Let S be the region bounded by the curves y = x³and y² = x. The curve y = 2|x| divides S into two regions of areas R₁ and R₂. If max {R₁,R₂} = R₂ , then $\frac{R_{2}}{R_{1}}$ is equal to ____ .

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answer is 19.

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Detailed Solution

$\begin{array}{l} S = \int_{0}^{1} \sqrt{x} - x^{3} \\ = {[\frac{2 x^{3 / 2}}{3} - \frac{x^{4}}{4}]}_{1}^{0} \\ R_{1} + R_{2} = \frac{5}{12} \\ R_{1} = \int_{0}^{1 / 4} (\sqrt{x} - 2 x) dx \\ = {[\frac{2 x^{3 / 2}}{3} - x^{2}]}_{0}^{1 / 4} = \frac{1}{48} \\ ∴ R_{2} = \frac{5}{12} - \frac{1}{48} = \frac{19}{48} \end{array}$
So, $\frac{R_{2}}{R_{1}} = 19$