Question

Calculate the area bounded by curve $y=x$ and x-axis from x = 1 to $\sqrt{5}$ .

Answer 1

We have to find the hatched areas shown in figure which is a trapezium
$A={\int }_1^{\sqrt{5}}ydx={\int }_1^{\sqrt{5}}xdx={\left[\frac{x^2}{2}\right]}_1^{\sqrt{5}}=\left(\frac{5}{2}\right)-\left(\frac{1}{2}\right)=2\text{units}$
We also know area of a trapezium is,
$A=\left(\frac{\left(\text{sum of parallel side}\right)}{2}\right)\times \left(\text{distance between parallel sides}\right)$
$A=\left(\frac{1+\sqrt{5}}{2}\right)(\sqrt{5}+1)=\frac{\left({\sqrt{5}}^2\right)-(1{)}^2}{2}=\frac{4}{2}=2\text{units}$