Question

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

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

Answer 1

The correct option is C $\frac{1}{6}$

$f^{\prime }\left(x\right)=2x+1$

$f\left(x\right)=x^2+x+c$

Curve passes through (1, 2)

$\Rightarrow 2=1+1+c$

$\Rightarrow c=0$

$f\left(x\right)=x^2+x=x(x+1)$

Curve passes through origin and (-1, 0)

$f(-0.5)=(-0.5{)}^2-0.5=0.25-0.5=-0.25$

Area $=\mid {\int }_{-1}^0(x^2+x)dx\mid =\mid [\frac{x^3}{3}+\frac{x^2}{2}{]}_{-1}^0=\frac{1}{6}$