Question

The area(in $\text{sq.units}$) included by the curve $x=2y-y^2$ and $y-$ axis is

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

Answer 1

The correct option is D $\frac{4}{3}$


Given $x=2y-y^2$ and $y-$axis
$\frac{dx}{dy}=2-2y=0\Rightarrow y=1$
$\Rightarrow$ curve bends at $y=1,$
$A=\underset{0}{\overset{2}{\int }}xdy=\underset{0}{\overset{2}{\int }}(2y-y^2)dy={[y^2-\frac{y^3}{3}]}_0^2=4-\frac{8}{3}=\frac{4}{3}$