cylindricalshell

The cylindrical shell method is a technique used in calculus to find the volume of a solid of revolution. It is particularly useful when revolving a region around an axis parallel to the axis of the integration variable. Imagine dividing the region into thin vertical or horizontal strips. When this strip is revolved around the axis of rotation, it forms a thin cylindrical shell.

The volume of a single cylindrical shell is approximated by the product of its circumference, its height, and its thickness. If the region is defined by a function $f(x)$ and is revolved around the y-axis, a vertical strip at position $x$ with width $dx$ has height $f(x)$. The radius of the cylindrical shell formed by revolving this strip is $x$, and its circumference is $2\pi x$. The volume of this infinitesimal shell is then $dV = (2\pi x) \cdot f(x) \cdot dx$.

To find the total volume of the solid, we integrate the volumes of all these infinitesimally thin cylindrical shells over the appropriate interval. For a region bounded by $y = f(x)$, the x-axis, and the lines $x=a$ and $x=b$, revolved around the y-axis, the volume $V$ is given by the integral: $V = \int_a^b 2\pi x f(x) dx$. If the axis of revolution is different, the radius term will change accordingly. Similarly, if horizontal strips are used and revolved around the x-axis, the integral would be in terms of $y$.