Find the area of the region bounded by the curve $y = x^2$ and the line $y = 4$.

$\frac{32}{3}$

The correct answer is Option (3) → $\frac{32}{3}$

Since the given curve represented by the equation $y = x^2$ is a parabola symmetrical about $y$-axis only, therefore, from given figure, the required area of the region $AOBA$ is given by

$2 \int\limits_{0}^{4} x \, dy = 2 \left( \begin{gathered} \text{area of the region } BONB \text{ bounded by curve, } y-\text{axis} \\ \text{and the lines } y = 0 \text{ and } y = 4 \end{gathered} \right) \text{}$

$= 2 \int\limits_{0}^{4} \sqrt{y} \, dy = 2 \times \frac{2}{3} \left[ y^{\frac{3}{2}} \right]_{0}^{4} = \frac{4}{3} \times 8 = \frac{32}{3}$