A parabola x = y² with 0 ≤ x ≤ 1 is shown in the figure. The volume of the solid of rotation obtained by rotating the shaded area by 360° around the x–axis is

The correct option is π/2.

To find the volume of the solid of rotation obtained by rotating the shaded area 360° around the x-axis, we use the disk method.

The boundary curve is given by the parabola equation:
$x=y^2$
for the interval:
$0\le x\le 1$

The formula for the volume $V$ of a solid rotated about the x-axis is given by:
$V={\int }_a^b$

Here, the limits of integration along the x-axis are from $a=0$ to $b=1$, as visible in the provided figure. Substituting these limits and the relation $y^2=x$ into the volume formula, we get:
$V={\int }_0^1\pi xdx$

Now, we integrate with respect to $x$:
$V=\pi {\left[\frac{x^2}{2}\right]}_0^1$

Evaluating the expression by substituting the upper limit 1 and the lower limit 0:
$V=\pi \left(\frac{1^2}{2}-\frac{0^2}{2}\right)$
$V=\pi \left(\frac{1}{2}-0\right)=\frac{\pi }{2}$

Therefore, the volume of the solid of rotation is $\frac{\pi }{2}$.