The area bounded between the curves \(y=a x^2\) and \(x=\mathrm{a} y^2(\mathrm{a}\gt0)\) is 1 sq. units, then the value of \(a\) is

The two curves intersect at \(O(0,0)\) and \(P\left(\frac{1}{\mathrm{a}}, \frac{1}{\mathrm{a}}\right)\).

According to the given condition,
\(\begin{aligned}
& \int_0^{\frac{1}{2}}\left(\sqrt{\frac{x}{\mathrm{a}}}-\mathrm{a} x^2\right) \mathrm{d} x=1 \\
& \Rightarrow\left[\frac{2}{3 \sqrt{\mathrm{a}}} x^{3 / 2}-\frac{\mathrm{a} x^3}{3}\right]_0^{1 / \mathrm{a}}=1 \\
& \Rightarrow \frac{2}{3 \sqrt{\mathrm{a}}} \times \frac{1}{\mathrm{a}^{3 / 2}}-\frac{\mathrm{a}}{3} \times \frac{1}{\mathrm{a}^3}=1 \\
& \Rightarrow \frac{2}{3 \mathrm{a}^2}-\frac{1}{3 \mathrm{a}^2}=1 \Rightarrow \frac{1}{3 \mathrm{a}^2}=1 \\
& \Rightarrow \mathrm{a}=\frac{1}{\sqrt{3}} \quad \ldots[\because \mathrm{a}\gt0]
\end{aligned}\)