Question

Find the area of the region bounded by the curves $y^2=9xandy=3x$.

Answer 1

We have, $y^2=9xandy=3x$

$\Rightarrow (3x{)}^2=9x$

$\Rightarrow 9x^2-9x=0$

$\Rightarrow 9x(x-1)=0$

$\Rightarrow x=1,0$

$\therefore Requiredarea,A={\int }_0^1\sqrt{9x}dx-{\int }_0^{1}3xdx=3{\int }_0^1{x}^{\frac{1}{2}}dx-3{\int }_0^{1}xdx={\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]}_0^1-3{\left[\frac{x^2}{2}\right]}_0^1=3(\frac{2}{3}-0)-3(\frac{1}{2}-0)=2-\frac{3}{2}=\frac{1}{2}squnits$