JEE Main · 2022 · 24 Jun 2022 · Shift 1 · Area Under Curves

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 ______.

Answer: 19

Solution

The curves

y = x^{3}

and

y^{2} = x

(i.e.,

y = x

for

x \geq 0

) intersect at

x = 0

and

x = 1

.

Total area

S = \int_{0}^{1} (x - x^{3}) d x = [\frac{2 x ^{3/2}}{3} - \frac{x ^{4}}{4}]_{0}^{1} = \frac{2}{3} - \frac{1}{4} = \frac{5}{12}

.

The line

y = 2 x

intersects

y = x

2 x = x

, giving

x = \frac{1}{4}

R_{1} = \int_{0}^{1/4} (x - 2 x) d x = [\frac{2 x ^{3/2}}{3} - x^{2}]_{0}^{1/4} = \frac{2}{3} \cdot \frac{1}{8} - \frac{1}{16} = \frac{1}{12} - \frac{1}{16} = \frac{1}{48}

R_{2} = S - R_{1} = \frac{5}{12} - \frac{1}{48} = \frac{20 - 1}{48} = \frac{19}{48}

\frac{R _{2}}{R _{1}} = \frac{19/48}{1/48} = 19

JEE Main · 2022 · 24 Jun 2022 · Shift 1 · Integer answer