35. Volume of Cylindrical Shells

Dated: 30-10-2024

Cylindrical Shells

We get this when we extend a washer, a disk with a hole in it, up.
Just like finding areas between 2 curves,

\[V = (\pi r_2^2 - \pi r_1^2) \cdot h\]

Here \(r_2\) is the radius of the outer circle and \(r_1\) is the radius of the inner circle.

\[V = \pi (r_2^2 - r_1^2) \cdot h\]

\[=\pi(r_2+r_1)(r_2-r_1)h\]

\[=2\pi\left(\frac{1}{2}(r_2+r_1)\right)h.(r_2-r_1)\]

The \(r_2 - r_1\) factor determines the thickness.

Let us say there is a region \(R\) bounded up by \(f(x)\), left and right by \(x = a\) and \(x = b\) respectively and below by x-axis.

We can rotate this region \(R\) around the y axis.
Then we divide the interval¹ into subintervals with points \(x_1, x_2, \ldots\)
This will give us a collection of solids similar to cylindrical shells but the upper surface will be defined according to \(f(x)\).

So the whole volume of the whole solid can be written as

\[V(S) = V(S_1) + V(S_2) + \ldots + V(S_n) = \sum_{i = 1}^n V(S_i)\]

Using the formula for volume of cylindrical shells, we can define volume of any arbitrary sub-solid as follows

\[V(S_i) = 2\pi \cdot f(x_i^*) \cdot \Delta x_i \cdot x_i^*\]

Here the \(x_i^*\) is the average radius which is good for approximation of height defined by \(f(x_i^*)\), when the thickness is very small, that is \(max(\Delta x_i) \to 0\)

\[V(S)=\lim_{\max(\Delta x_i)\to 0}\sum_{i=1}^{n}2\pi x_i^*f(x_i^*)\Delta x_i=\int_{a}^{b}2\pi xf(x)dx\]

35. Volume of Cylindrical Shells

Cylindrical Shells

References