Area of the region (in sq units) bounded by the curves $y=\sqrt{x}, x=\sqrt{y}$ and the lines $x=1$, $x=4$, is

• A

  $\frac{8}{3}$
• Correct answer
  B

  $\frac{49}{3}$
• C

  $\frac{16}{3}$
• D

  $\frac{14}{3}$

Solution

We have, $$ y=\sqrt{x}, x=\sqrt{y}, x=1, x=4 $$ The graph of curves are
Area of shaded region $$ \begin{aligned} & =\int_1^4\left(x^2-\sqrt{x}\right) d x=\left[\frac{x^3}{3}-\frac{2}{3}(x)^{3 / 2}\right]_1^4 \\ & =\left(\frac{64}{3}-\frac{16}{3}\right)-\left(\frac{1}{3}-\frac{2}{3}\right) \\ & =\frac{64-16-1+2}{3}=\frac{49}{3} \end{aligned} $$

Hence, the correct answer is option: 'B'.