Question

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

Answer 1

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

We have
$\frac{dy}{dx}=2x+1$
$y=x^2+x+c$
Given that the curve passes through $(1,2)\Rightarrow c=0$
$\therefore y=x^2+x$
Required area
$=\underset{0}{\overset{1}{\int }}(x^2+x)dx={[\frac{x^3}{3}+\frac{x^2}{2}]}_0^1=\frac{5}{6}$