Question

If the area of the closed figure bounded by the following curves $y=2(1-x),y=1-$ $x^2$, $x=0$ is $\frac{1}{k}$. Find k

Answer 1

Area of the shaded region $=$ area of $\Delta ABC-$ area of $DBC$

$=\frac{1}{2}\times 2\times 1-{\int }_0^{1}y{d}_{DC}$

$=1-{\int }_0^{1}y{d}_{DC}$

$=1-{\int }_0^1(1-x^2)dx$

$=1-{[x-\frac{x^3}{3}]}_0^1$

$=1-\frac{2}{3}=\frac{1}{3}=\frac{1}{k}$

$\therefore k=3$