Find the area of the region bounded by the curve $y^2 = x$ and the lines $x=1$, $x=4$ and the x-axis in the first quadrant.

The correct answer is $\dfrac{14}{3}$

Required Area ABCD = $\int_a^b y$ $dx$

= $\int_1^4 \sqrt{x}$ $dx$

= $\int_1^4 x^{\frac{1}{2}}$ $dx$


= $\left[ \dfrac{x^{\frac{1}{2}+ 1}}{\frac{1}{2}+ 1} \right]_1^4$


= $\left[ \dfrac{x^{\frac{3}{2}}}{\frac{3}{2}} \right]_1^4$


= $\dfrac{2}{3} \left[ 4^{\frac{3}{2}} - 1^{\frac{3}{2}} \right]$

= $\dfrac{2}{3} \left[ 8 - 1 \right]$

= $\dfrac{14}{3}$