
Q1 Find the area of the region bounded by the curve y^{2} = x and the lines x = 1, x = 4 and the xaxis.
Ans: The area of the region bounded by the curve, y^{2} = x, the lines, x = 1 and x = 4, and the xaxis is the area ABCD.
\begin{align}Area \;of\; ABCD = \int_{1}^{4} y.dx \end{align}
\begin{align} = \int_{1}^{4} \sqrt{x}.dx \end{align}
\begin{align} =\left[\frac{x^\frac{3}{2}}{\frac{3}{2}}\right]_1^4 \end{align}
\begin{align} =\frac{2}{3}\left[(4)^\frac{3}{2}  (1)^{\frac{3}{2}}\right] \end{align}
\begin{align} =\frac{2}{3}\left[8 1\right] \end{align}
\begin{align} =\frac{14}{3}\; Units \end{align}