- When the arc of a curve of \(y = f(x)\) on the interval \(a \le x \le b\) is rotated about the \(x\)-axis, the volume of the solid of revolution is given by \(V = \pi \int\limits_a^b {{{\left( {f\left( x \right)} \right)}^2}} dx\)
- When the arc of \(x = g(y)\) a curve of on the interval \(c \le x \le d\) is rotated about the \(y\)-axis, the volume of the solid of revolution is given by \(V = \pi \int\limits_c^d {{{\left( {g\left( y \right)} \right)}^2}} dy\)
For Example: Part of the parabola \(y = {x^2}\) between \(x = 1\) and \(x = 2\) is rotated about the \(x\)-axis. Find the volume so formed.
\[\begin{align*}V = \pi \int\limits_1^2 {{{\left( {{x^2}} \right)}^2}} dx &= \pi \int\limits_1^2 {{x^4}} dx\\ &= \dfrac{\pi }{5}\left[ {{x^5}} \right]_1^2 = \dfrac{{31\pi }}{5}\,\,{u^3}\end{align*}\]
For Example: Part of the parabola \(y = {x^2}\) between \(x = 1\) and \(x = 2\) is rotated about the \(y\)-axis. Find the volume so formed.
\[\begin{align*}V &= \pi \int\limits_1^4 {y\,} dy\\ &= \dfrac{\pi }{2}\left[ {{y^2}} \right]_1^4 = \dfrac{{15\pi }}{2}\,\,{u^3}\end{align*}\]