Question

Draw a rough sketch of the curve y = ${\displaystyle \sqrt{x-1}}$ in the interval [1, 5]. Find the area under the curve and between the lines x = 1 and x = 5.

Answer 1

Here, we have y = ${\displaystyle \sqrt{x-1}}$

⇒ y² = x – 1   ......(Parabola)

Area of the required region

= ${\displaystyle {\int }_1^5\sqrt{x-1} \text{d}x}$

= ${\displaystyle \frac{2}{3}{\left[{(x-1)}^{\frac{3}{2}}\right]}_1^5}$

= ${\displaystyle \frac{2}{3}[{(5-1)}^{\frac{3}{2}}-0]}$

= ${\displaystyle \frac{2}{3}\times {\left(4\right)}^{\frac{2}{3}}}$

= ${\displaystyle \frac{2}{3}\times 8}$

= ${\displaystyle \frac{16}{3}}$ sq.units