Common misconceptions about volume of revolution

NOTE:

1. Volume obtained is NOT \(\pi \int_{a}^{b} (\mathrm{f}(x) - g(x))^2 \, \mathrm{d}x\)
2. Similarly, volume of solid obtained by rotating the region bounded by \(x = \mathrm{f}(y)\), \(x = \mathrm{g}(y)\) and the lines \(y = c\) and \(y = d\) about the \(y\)-axis through \(360{}^\circ\) is \[\pi \int_{c}^{d} ([\mathrm{f}(y)]^2 - [\mathrm{g}(y)]^2) \, \mathrm{d}y\]