Question

Find the area of region bounded by y^2 =4a^2 (x-1) and the lines x=1 and y=4a

Answer 1

$$\begin{gathered} \mathrm{Given}, \\ {\mathrm{y}}^2=4{\mathrm{a}}^2\left(\mathrm{x}-1\right) \\ \mathrm{y}=2\mathrm{a}\sqrt{\mathrm{x}-1} \\ \mathrm{Required}\mathrm{area}\mathrm{will}\mathrm{be} \\ ={\int }_1^{4\mathrm{a}}\mathrm{ydx}=2\mathrm{a}{\int }_1^{4\mathrm{a}}\sqrt{\mathrm{x}-1}\mathrm{dx}=2\mathrm{a}\times \frac{2}{3}{\left[{\left(\mathrm{x}-1\right)}^{\frac{3}{2}}\right]}_1^{4\mathrm{a}} \\ =\frac{4\mathrm{a}}{3}{\left[4\mathrm{a}\right]}^{\frac{3}{2}} \\ =\frac{{\left(4a\right)}^{{\displaystyle \frac{5}{2}}}}{3}\mathrm{square}\mathrm{unit} \end{gathered}$$