Question

The parabola $y^2=4x$ and $x^2=4y$ divide the square region bounded by the lines $x=4,y=4$ and the coordinate axes. If $S_1,S_2$ and $S_3$ are respectively the areas of these parts numbered from top-to-bottom, then $S_1:S_2:S_3$ is

• A. $1:1:1$
• B. $2:1:2$
• C. $1:2:3$
• D. $1:2:1$

Answer 1

The correct option is A $1:1:1$

$S_1=S_3={\int }_0^4\frac{x^2}{4}dx$
$={\left[\frac{x^3}{12}\right]}_0^4=[\frac{64}{12}-0]=\frac{16}{3}$
Now, $S_2={\int }_0^4(\sqrt{4x}-\frac{x^2}{4})dx$
$={[2x^{3/2}\cdot \frac{2}{3}-\frac{x^3}{12}]}_0^4=\frac{16}{3}$
$\therefore S_1:S_2:S_3=\frac{16}{3}:\frac{16}{3}:\frac{16}{3}=1:1:1$