Question

The area bounded by $x=-4y^2$ and $x-1=5y^2$ is

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

Answer 1

Answer: (C)

We have $x=-4y^2...\left(i\right)$, a parabola with vertex $\left(0,0\right)$
and $-5y^2=x-1...\left(ii\right)$, a parabola with vertex $\left(1/5,0\right)$
Solving $\left(i\right)$ and $\left(ii\right)$, we get $y=\pm 1$

Area $={\int }_{-1}^1\left(1-5y^2-\left(-4y^2\right)\right)dy<br/>={\int }_{-1}^1\left(1-y^2\right)dy$
$=2{\int }_0^1\left(1-y^2\right)dy$
$=2{\left[y-\frac{y^3}{3}\right]}_0^1$
$=2\left(1-\frac{1}{3}\right)$
$=\frac{4}{3}$ sq. units