Question

The slope of the tangent to the curve $y=f\left(x\right)$ at $(x,f(x\left)\right)$ is $2x+1$. lf 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. $\frac{5}{6}$
• B. $\frac{6}{5}$
• C. $\frac{1}{6}$
• D. $6$

Answer 1

Answer: (A)

$\frac{5}{6}$

We have $\frac{dy}{dx}=2x+1$
Integrating both sides w.r.t $x$, we get
$y=x^2+x+c$
Since the curve passes through $(1,2)$
$\Rightarrow 2=1+1+c\Rightarrow c=0$
Therefore $y=x^2+x$
Hence required area $={\int }_0^1(x^2+x)dx={[\frac{x^3}{3}+\frac{x^2}{2}]}_0^1=\frac{5}{6}$