Calculus/Volume

Volumes

In this section we will learn how to find the volume of a shape. The procedure is very similar to calculating the Area. The basic procedure is:

• Partition the shape in
• Calculate basal area of every partition
• Multiply by height of the partition
• Sum up all the volumes

So, given a function $f(x)$ that gives us the basal area at a given height x, we can write it up as follows:

$\sum _{i=1}^{n}f(x_i)\mathrm{\Delta }x$ Now limit it to infinity:

$\underset{n\to \mathrm{\infty }}{\lim }\sum _{i=1}^{n}f(x_i)\mathrm{\Delta }x$ This is a Riemann's Sum, so we can rewrite it as:

$\underset{a}{\overset{b}{\int }}f(x)dx$

Examples

Calculate the volume of a square pyramid of base b and height h.

The basal shape is a square, and depends on the height x at which it is taken. For simplicity, we will consider an inverted pyramid, so that $f(x)={\left(\frac{b}{h}x\right)}^2$. We integrate in the proper range (0 to h):

$\underset{0}{\overset{h}{\int }}{\left(\frac{b}{h}x\right)}^{2}dx={\left(\frac{b}{h}\right)}^2\underset{0}{\overset{h}{\int }}{x}^{2}dx={\left(\frac{b}{h}\right)}^2\frac{h^3}{3}=\frac{b^2\cdot h}{3}$