Volumes of Revolution

One the "cooler" application of calculus is the idea behind volumes of revolution. The idea is to take a function rotate it (for SL we only rotate around the x-axis) to to create a 3D shape. For example the arbitrary function f(x) as shown below: This function is rotated, imagine rotating a 2D shape bounded by the function and the two axes, to create a 3D vase like structure. We can then imagine slicing the structure into VERY thin, actually infinitesimally thin, discs. The volume of a disc (or cylinder) can be calculated with the equation:

(1)
\begin{align} V_{slice}=V_{cylinder}=\pi r^2 h \end{align}

In the case of our very thin discs, the height or thickness of it is dx. The radius is simply the function value. Giving a volume of:

(2)
\begin{align} V_{slice}=\pi (f(x))^2 dx = \pi y^2 dx \end{align}

One way of thinking about an integral is to think of it as a tool to add or sum up things. When we calculated the area under a curve we made the argument of adding up all of the infinitesimally thin rectangles… We can use an integral to add up the volume of all the infinitesimally thin discs! Giving an equation for the volume as:

(3)
\begin{align} V_{total}=\int_a^b \pi y^2 dx \end{align}

Where a and b are the ends of the function. This last equation is the equation in the IB formula booklet.

#### Example 1

Find the volume of the structure created by rotating $f(x)=\frac{1}{4} x^2$ for $0\leq x \leq 2$.