Question

The area bounded by the curve $y=x^{2}\left(x - 1\right)^{2}$ with the $x$ -axis is $k$ sq. units, then the value of $60k$ is equal to

• A. $1$
• B. $2$
• C. $\frac{1}{2}$
• D. $\frac{1}{4}$

Answer 1

Answer: (B)

The required area is $A=\displaystyle \int _{0}^{1} x^{2} \left(x - 1\right)^{2} d x$
or $A=\displaystyle \int _{0}^{1} \left(x^{2} - x\right)^{2} d x=\displaystyle \int _{0}^{1} \left(x^{4} + x^{2} - 2 x^{3}\right)dx$
$=\left[\frac{x^{5}}{5} + \frac{x^{3}}{3} - \frac{2 x^{4}}{4}\right]_{0}^{1}$
$=\frac{1}{5}+\frac{1}{3}-\frac{2}{4}=\frac{1}{30}=k$
$\therefore 60k=2$