Question

Let $A_1,A_2$ denote the area bounded by the curve $y=x|x|,x$ axis, the ordinates $x=-1,x=1$ and the area enclosed between the curves $y^2=x$, $y=|x|.$ Find $\frac{A_1}{A_2}$.

Answer 1

Answer: 4

$A_1=2\underset{0}{\overset{1}{\int }}{x}^{2}dx=2{\left(\frac{x^3}{3}\right)}_0^1=\frac{2}{3}$
$A_2=\underset{0}{\overset{1}{\int }}(\sqrt{x}-x)dx={\left(\frac{2}{3}{x}^{\frac{3}{2}}-\frac{x^2}{2}\right)}_0^1=\frac{2}{3}-\frac{1}{2}$
$=\frac{4-3}{6}=\frac{1}{6}$
$\Rightarrow \frac{A_1}{A_2}=\frac{\left(\frac{2}{3}\right)}{\left(\frac{1}{6}\right)}=\frac{6\times 2}{3}=4$