The area of the plane region is ${\int }_0^1{x}^2 \mathit{\ⅆ}x\=\frac{1}{3}$.

Since the rotation is about the $y$-axis, the radius of rotation for the centroid is   $\stackrel{\&conjugate0;}{y}\=\frac{1}{1\/3}{\int }_0^1\frac{1}{2}{\left(x^2\right)}^2 \mathit{\ⅆ}x\=\frac{3}{10}$.  

Hence, the centroid traverses a distance of $2 \mathrm{\π}\left(3\/10\right)\=3 \mathrm{\π}\/5$, so by the first theorem of Pappus, the volume of the solid of revolution is $3 \mathrm{\π}\/5\cdot \left(1\/3\right)\=\mathrm{\π}\/5$, as found in Example 5.2.1.