Area of element \(d A =2 \pi rd \ell==2 \pi r \frac{ dh }{\cos \theta}\)

mass of element \(dm =\frac{ M }{\pi R \sqrt{ H ^{2}+ R ^{2}}} \times \frac{2 \pi rdh }{\cos \theta}\)

\(dm =\frac{2 Mh \tan \theta dh }{ R \sqrt{ H ^{2}+ R ^{2}} \cos \theta} \quad\) (here \(\left. r = h \tan \theta\right)\)

\(I =\int( dm ) r ^{2}=\int \frac{ h ^{2} \tan ^{2} \theta}{\cos \theta}\left(\frac{2 m }{ R } \frac{ h \tan \theta}{\sqrt{ R ^{2}+ H ^{2}}}\right) dh\)

\(=\frac{2 M }{\cos \theta R } \frac{\tan ^{3} \theta}{\sqrt{ R ^{2}+ H ^{2}}}\)

\(\int_{0}^{ H } h ^{3} dh =\frac{ MR ^{2} H ^{4}}{2 RH ^{3} \sqrt{ R ^{2}+ H ^{2}} \cos \theta}\)

\(=\frac{ MR ^{2} H \sqrt{ R ^{2}+ H ^{2}}}{2 \sqrt{ R ^{2}+ H ^{2}} \times H }\)

\(=\frac{ MR ^{2}}{2}\)