Question

Let f(x)={x2;x<0x;x≥0f(x)={x2;x<0x;x≥0 is equal to:

• A. $\frac{1}{2}$
• B. $-\frac{1}{2}$
• C. $0$
• D. none of these.

Answer 1

The correct option is A $\frac{1}{2}$

If $a>0,$ then
$\therefore$ Required Area$={\int }_{-3a}^0{x}^{2}dx+{\int }_0^{3a}x.dx$
$={\int }_0^{3a}(x^2+x)dx$
$={[\frac{x^3}{3}+\frac{x^2}{2}]}_0^{3a}$
$=\frac{27a^3}{3}+\frac{9a^2}{2}=\frac{9a}{2}$ (given)
$=a^2+\frac{a}{2}=\frac{1}{2}$ $(\because a\ne 0)$
Multiply the above equation by $2$ we get
$\Rightarrow 2a^2+a-1=0$
we get, $a=\frac{1}{2},a=-1$
$\therefore a=\frac{1}{2},a\ne 1$