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Q.
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
Application of Integrals
Solution:
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
$=\int\limits_1^5 \sqrt{x-1} d x$
$ =\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
