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BITSAT 2021

Exam: BITSAT
Year: 2021
Subject: Mathematics
Chapter: Area Under Curves

BITSAT 2021 · Mathematics · Area Under Curves

The area bounded by the curves x+2 y^ 2 =0 and x+3 y^ 2 =1 is

The area bounded by the curves

x+2 y^{2}=0

and

x+3 y^{2}=1

is

The area bounded by the curves x+2 y^ 2 =0 and x+3 y^ 2 =1 is

Options

A1 sq. unit
B $\frac{1}{3}$ sq. units
C $\frac{2}{3}$ sq. units
D $\frac{4}{3}$ sq. units

Correct answer

D. $\frac{4}{3}$ sq. units

Step-by-step solution

We have,

x+2 y^{2}=0\Rightarrow y^{2}=-\frac{x}{2} \ldots \text{ (i) }

a parabola with vertex

(0,0)

and

x+3 y^{2}=1\Rightarrow y^{2}=\frac{1-x}{3}=-\left(\frac{x-1}{3}\right) \ldots

(ii), a parabola with vertex

(1,0)

Solving (i) and (ii), we get

y=\pm 1

The area bounded by the curves x+2 y^ 2 =0 and x+3 y^ 2 =1 is

\mathrm{A}=\int_{-1}^{1}\left(\left(1-3 y^{2}\right)-\left(-2 y^{2}\right)\right) d y=2 \int_{0}^{1}\left(1-y^{2}\right) d y=2\left[y-\frac{y^{3}}{3}\right]_{0}^{1}=\frac{4}{3}

sq.units

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