Question

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

• A. $1$
• B. $\frac{1}{2}$
• C. $2$
• D. $\frac{\sqrt{5}}{2}$

Answer 1

The correct option is C $2$

We have to find the shaded area shown in figure which is a trapezium.


$A={\int }_1^{\sqrt{5}}ydx={\int }_1^{\sqrt{5}}xdx={\left[\frac{x^2}{2}\right]}_1^{\sqrt{5}}=\left(\frac{5}{2}\right)-\left(\frac{1}{2}\right)=2\text{units}$
Alternative: We know area of a trapezium
$A=\left(\frac{\text{Sum of parallel sides}}{2}\right)\times$ (distance between parallel sides)
$A=\left(\frac{1+\sqrt{5}}{2}\right)(\sqrt{5}-1)=\frac{(\sqrt{5}{)}^2-(1{)}^2}{2}=\frac{4}{2}=2\text{units}$