Question

Find the area included between the two curves $y^2=9x$ and $x^2=9y$

Answer 1

Parabola $y^2=9x$ and $x^2=9y$ intersect at $(0,0)$ and $(9,9)$

Area of shaded part $OABC$

$=$ area $OABL-$ area $OCBL$

$={\int }_0^9\sqrt{9x}dx-{\int }_0^9\frac{x^2}{9}dx$

$=3{\int }_0^9\sqrt{x}dx-\frac{1}{9}{\int }_0^9{x}^{2}dx$

$=3{\left[\frac{2}{3}{x}^{3/2}\right]}_0^9-\frac{1}{9}{\left[\frac{x^3}{3}\right]}_0^9$

$=3\times \frac{2}{3}[9^{3/2}-0]-\frac{1}{9}[\frac{9^3}{3}-0]$

$=54-27=27sq.unit$