1. volume

How to find the solid generated by revolving a loop of the curve $r^2=a^2sin2 \theta$ about the initial line.

Here I used the fact that the volume

$V=\frac{2}{3}\int(f(\theta))^3sin\theta d\theta.$

But I couldn't get the answer. The answer should be $\frac{{\pi}^2 a^3}{8}$

2. Did you try the Theorem of Pappus?. That may be a good choice here.

Find the area of the region multiplied by the distance traveled by the centroid, then multiply by 2Pi