Question

The slope of the tangent to the curve $y=f\left(x\right)$ at $(x,f\left(x\right))$ is $2x+1$.If the curve passes through the point $(1,2)$ then the area of the region bounded by the curve, the $x-$axis and the line $x=1$ is

• A. $\frac{5}{6}$.sq.unit
• B. $\frac{6}{5}$.sq.unit
• C. $\frac{1}{6}$.sq.unit
• D. $6$.sq.unit

Answer 1

The correct option is A $\frac{5}{6}$.sq.unit

We have,$\frac{dy}{dx}=2x+1$
$\Rightarrow y=x^2+x+c,$ it passes through $(1,2)$
$\therefore c=0$
Then, $y=x^2+x$
$\therefore$Required area$={\int }_0^1(x^2+x)dx$
$={[\frac{x^3}{3}+\frac{x^2}{2}]}_0^1$
$=[(\frac{1}{3}-0)+(\frac{1}{2}-0)]$
$=\frac{5}{6}$.sq.unit.