Question

The area of the region bounded by the curve $y = x^3$, its tangent at $(1, 1)$ and $x-axis$ is

• A. $\frac{1}{12}$ sq unit
• B. $\frac{1}{16}$ sq unit
• C. $\frac{2}{17}$ sq unit
• D. $\frac{2}{15}$ sq unit

Answer 1

Answer: (A)

$\frac{1}{12}$ sq unit

Answer 2

We have, $y=x^{3}$ and $A(1,1)$
$\therefore \frac{d y}{d x}=3 x^{2}$ ...(i)
On putting $x=1$ in E (i), we get
$\frac{d y}{d x}=3(1)^{2}=3$
$\therefore$ Equation of tangent at $A(1,1)$ is
$y-1=3(x-1) \Rightarrow y=3 x-2$

$\therefore$ Required area
$=\int\limits_{0}^{1} x^{3} d x-\int\limits_{2 / 3}^{1}(3 x-2) d x$
$=\left[\frac{x^{4}}{4}\right]_{0}^{1}-\left[\frac{3 x^{2}}{2}-2 x\right]_{2 / 3}^{1}$
$=\frac{1}{4}-\left[\left(\frac{3}{2}-2\right)-\left(\frac{2}{3}-\frac{4}{3}\right)\right]$
$=\frac{1}{12}$ sq unit