Question

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

• A. $\frac{1}{12}$
• B. $\frac{1}{6}$
• C. $\frac{2}{17}$
• D. $\frac{2}{15}$

Answer 1

The correct option is A $\frac{1}{12}$

$y=x^3\Rightarrow \frac{dy}{dx}{|}_{(1,1)}=3x^2=3$
$\therefore$The tangent at $A$ is,
$y=3x-2$



$\text{Area}=\underset{0}{\overset{1}{\int }}{x}^{3}dx-\text{Area}\left(△ABC\right)$
$=\frac{1}{4}-\frac{1}{2}\times \frac{1}{3}\times 1$
$=\frac{1}{12}$