Question

The area of the region bounded by the curve x = y², y-axis and the lines y = 3 and y = 4 is _____________.

Answer 1

To find: area of the region bounded by the curve x = y², y-axis and the lines y = 3 and y = 4



The required area of the region ABCDA = ${\int }_3^{4}xdy$
Thus,
$$\begin{gathered} \mathrm{Required}\mathrm{area}={\int }_3^{4}xdy \\ ={\int }_3^4\left(y^2\right)dy \\ ={\left(\frac{y^3}{3}\right)}_3^4 \\ =\left(\frac{4^3}{3}\right)-\left(\frac{3^3}{3}\right) \\ =\left(\frac{64}{3}\right)-\left(\frac{27}{3}\right) \\ =\frac{64-27}{3} \\ =\frac{37}{3} \\ \mathrm{Thus},\mathrm{Area}=\frac{37}{3}\mathrm{sq}.\mathrm{units} \end{gathered}$$


Hence, the area of the region bounded by the curve x = y², y-axis and the lines y = 3 and y = 4 is $\underline{\frac{37}{3}\mathrm{sq}.\mathrm{units}}.$