Question

The area bounded by the curves $y=(x-1{)}^2$, $y=(x+1{)}^2$ and $y=\frac{1}{4}$ is

• A. $(a)\frac{1}{3}$ sq units
• B. $(b)\frac{2}{3}$ sq units
• C. $(c)\frac{1}{4}$ sq units
• D. $(d)\frac{1}{5}$ sq units

Answer 1

Answer: (A)

The curves $y=(x-1{)}^2$, $y=(x+1{)}^2$ and $y=\frac{1}{4}$ are shown as
where, points of intersection are
$(x-1{)}^2=\frac{1}{4}\Rightarrow x=\frac{1}{2}$
and $(x+1{)}^2=\frac{1}{4}\Rightarrow x=-\frac{1}{2}$
i.e $Q(\frac{1}{2},\frac{1}{4})andR(-\frac{1}{2},\frac{1}{4})$
$\therefore Requiredarea=2\underset{0}{\overset{1/2}{\int }}[(x-1{)}^2-\frac{1}{4}]dx$
$=2[\frac{(x-1{)}^3}{3}-\frac{1}{4}x{]}_0^{1/2}$
$=2[-\frac{1}{8.3}-\frac{1}{8}-(-\frac{1}{3}-0\left)\right]=\frac{8}{24}=\frac{1}{3}squnits$