Calculate the area under the curve $y = 2\sqrt{x}$ included between the lines $x = 0$ and $x = 1$.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → $\frac{4}{3}$ square units
We have,
$y = 2\sqrt{x} \dots(i)$
$x = 0 \dots(ii)$
and $x = 1 \dots(iii)$
On solving Eqs. (i), (ii) and (iii), we get
$x = 0 \Rightarrow y = 0$ and $x = 1 \Rightarrow y = 2$
$∴$ Intersecting points are $(0, 0)$ and $(1, 2)$.
$∴$ Area of shaded region $= \int\limits_{0}^{1} (2\sqrt{x}) \, dx$
$= 2 \cdot \left[ \frac{x^{3/2}}{3}\cdot 2 \right]_{0}^{1}$
$= 2 \left( \frac{2}{3} \cdot 1 - 0 \right) = \frac{4}{3} \text{ sq. units}$