Question

Find the area of the region bounded by the two parabolas $x^2=4y$ and $y^2=4x$. (Draw the figure in answer-book)

Answer 1

Intersecting points of the two given curves: $y^{=}4x$ and $x^2=4y$ is $(0,0)$ and $(4,4)$

Therefore, the area of the region bounded by these curves:

$A={\int }_0^4\sqrt{4x}-\frac{x^2}{4}$

$\Rightarrow A={\left[\frac{4}{3}{x}^{3/2}\right]}_0^4-{\left[\frac{x^3}{12}\right]}_0^4$

$\Rightarrow A=\frac{32}{3}-\frac{64}{12}$

$\Rightarrow A=\frac{64}{12}=\frac{16}{3}sq.units$