Question

Find the area bounded by the parabola $y^2=5x$ and the line $y=x$.

Answer 1

$y_1=x{y}_2=\sqrt{5x}$

$A_2={\int }_0^5[x-\sqrt{5x}]dx$

$A=|{\int }_0^5[x-\sqrt{5}\sqrt{x}]dx|$

$A={[\frac{x^2}{2}-\sqrt{5}\frac{x^3/2}{3/2}]}_0^5$

$A=\frac{25}{2}-\sqrt{5}\times \frac{2}{3}\times (5{)}^{3/2}-0+0$

$A=\frac{25}{2}-5^{3/2+1/2}\times \frac{2}{3}=\frac{25}{2}-\frac{50}{3}$

$A=\left|\frac{75-100}{6}\right|=A=\frac{25}{6}$