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 re- gion bounded by the curve, the $x$-axis and the line $x=1$ is:

• A. $\frac{2}{3}$
• B. $\frac{1}{3}$
• C. $\frac{1}{6}$
• D. $\frac{5}{6}$

Answer 1

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

$f^{\prime }\left(x\right)=2x+1$
$\int f^{\prime }\left(x\right)dx=\int (2x+1)dx$
$f\left(x\right)=x^2+x+c$ passes through (1,2)
$2=2+c\to c=0$
So, $f\left(x\right)=x^2+x$
$\stackrel{1}{\underset{0}{\int }}(x^2+x)dx$
$1/3+1/2=5/6squnits.$