Question

The area (in sq. units) bounded by the curve $y=x^2+2x+1$ and the tangent to it at $(1,4)$ and the $Y$ - axis is

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. 1
• D. $\frac{7}{3}$

Answer 1

Answer: (A)

Given, $y=x^2+2x+1$
Differentiating w.r.t. $x$, we get
${\therefore \frac{dy}{dx}|}_{P(1,4)}=2+2=4$
$\therefore$ Equation of tangent at $P(1,4)$
$y-4=4(x-1)$
$y-4=4x-4$
$y=4x$

Required area $=\underset{0}{\overset{1}{\int }}ydx-\frac{1}{2}\times OA\times AP$
$=\underset{0}{\overset{1}{\int }}\left(x^2+2x+1\right)dx-\frac{1}{2}\times 1\times 4$
$={\left(\frac{x^3}{3}+x^2+x\right)}_0^1-2=\left(\frac{1}{3}+1+1-0\right)-2$
$=\frac{7}{3}-2=\frac{1}{3}$