Volumes generated by rotating curves about the x-axis

1 Volumes generated by rotating curves about the x-axis

Figure 8 shows a graph of the function $y = 2 x$ for $x$ between 0 and 3.

Figure 8 :

{ A graph of the function $y=2x$, for 0 less than or equal to x less than or equal to 3}

Imagine rotating the line $y = 2 x$ by one complete revolution ( $36 0^{0}$ or $2 π$ radians) around the $x$ -axis. The surface so formed is the surface of a cone as shown in Figure 9. Such a three-dimensional shape is known as a solid of revolution . We now discuss how to obtain the volumes of such solids of revolution.

Figure 9 :

{ When the line $y=2x$ is rotated around the axis, a solid is generated}

Task!

Find the volume of the cone generated by rotating $y = 2 x$ , for $0 \leq x \leq 3$ , around the $x$ -axis, as shown in Figure 9.

In order to find the volume of this solid we assume that it is composed of lots of thin circular discs all aligned perpendicular to the $x$ -axis, such as that shown in the diagram. From the diagram below we note that a typical disc has radius $y$ , which in this case equals $2 x$ , and thickness $δ x$ .

{The cone is divided into a number of thin circular discs.}

The volume of a circular disc is the circular area multiplied by the thickness.

Write down an expression for the volume of this typical disc:

$π {(2 x)}^{2} δ x = 4 π x^{2} δ x$

To find the total volume we must sum the contributions from all discs and find the limit of this sum as the number of discs tends to infinity and $δ x$ tends to zero. That is

$lim_{δ x \to 0} \sum_{x = 0}^{x = 3} 4 π x^{2} δ x$

This is the definition of a definite integral. Write down the corresponding integral:

$\int_{0}^{3} 4 π x^{2} d x$

Find the required volume by performing the integration:

${[\frac{4 π x^{3}}{3}]}_{0}^{3} = 36 π$

Task!

A graph of the function $y = x^{2}$ for $x$ between 0 and 4 is shown in the diagram. The graph is rotated around the $x$ -axis to produce the solid shown. Find its volume.

{The solid of revolution is divided into a number of thin circular discs.}

As in the previous Task, the solid is considered to be composed of lots of circular discs of radius $y$ , (which in this example is equal to $x^{2}$ ), and thickness $δ x$ .

Write down the volume of each disc:

$π {(x^{2})}^{2} δ x = π x^{4} δ x$

Write down the expression which represents summing the volumes of all such discs:

$\sum_{x = 0}^{x = 4} π x^{4} δ x$

Write down the integral which results from taking the limit of the sum as $δ x \to 0$ :

$\int_{0}^{4} π x^{4} d x$

Perform the integration to find the volume of the solid:

$\frac{4^{5} π}{5} = 204.8 π$

Task!

In general, suppose the graph of $y (x)$ between $x = a$ and $x = b$ is rotated about the $x$ -axis, and the solid so formed is considered to be composed of lots of circular discs of thickness $δ x$ .

Write down an expression for the radius of a typical disc:

$y$

Write down an expression for the volume of a typical disc:

$π y^{2} δ x$

The total volume is found by summing these individual volumes and taking the limit as $δ x$ tends to zero:

$lim_{δ x \to 0} \sum_{x = a}^{x = b} π y^{2} δ x$

Write down the definite integral which this sum defines:

$\int_{a}^{b} π y^{2} d x$

Key Point 5

If the graph of $y (x)$ , between $x = a$ and $x = b$ , is rotated about the $x$ -axis the volume of the solid formed is

\int_{a}^{b} π y^{2} d x

Exercises

Find the volume of the solid formed when that part of the curve between $y = x^{2}$ between $x = 1$ and $x = 2$ is rotated about the $x$ -axis.
The parabola $y^{2} = 4 x$ for $0 \leq x \leq 1$ , is rotated around the $x$ -axis. Find the volume of the solid formed.

1. $31 π ∕ 5$ , 2. $2 π$ .

1 Volumes generated by rotating curves about the x-axis

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Key Point 5

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