Question

If we draw a rough sketch of the curve $y=\sqrt{x-1}$ in the interval $[1,5]$, then the area under the curve and between the lines $x=1$ and $x=5$ is

• A. $\frac{16}{9}$ sq units
• B. $\frac{\overline{8}}{3}$ sq units
• C. $\frac{16}{3}$ sq units
• D. None of the above

Answer 1

Answer: (C)

We have,
Equation of the curve
$y=\sqrt{x-1}$
On squaring both sides, we get
$y^2=x-1$
Now, sketch the graph of given curve.

$\therefore$ Area of the shaded region
$=\underset{1}{\overset{5}{\int }}\sqrt{x-1}dx$
$={\left[\frac{(x-1{)}^{3/2}}{3/2}\right]}_1^5=\frac{2}{3}{\left[(x-1{)}^{3/2}\right]}_1^5$
$=\frac{2}{3}\left[(4{)}^{3/2}-0\right]$
$=\frac{2}{3}\times 8$
$=\frac{16}{3}$ sq units