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Revision:Volumes of Revolution

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Volumes of Revolution

Contents

Volumes of Revolution

Sometimes it is necessary to calculate the volume of a solid - this can be obtained by rotating a curve around the x-axis. This is done through a straightforward integration technique.


Rotation about the x-axis

Suppose we have a curve y=f(x).

Image:Function.jpg

Imagine you could rotate this curve through 360° between the points x=a and x=b - this would map out a solid as it is rotated. If the curve were a circle we would get a sphere, if it were a straight line we would get a cone and so on. (The formulae for the volume of both spheres and cones can be proven using this method - as shown below).

Image:Function2.jpg

In order to get a numerical answer or algabraic expression use this rule:

If y is a function of x, then the volume of the solid obtained by rotating the portion of the curve between x=a and x=b is:

\displaystyle \text{V}=\pi\int^{\text{b}}_{\text{a}}y^2\text{d}x


Examples

Volume of a Sphere

The equation: x^2+y^2=\text{r}^2 represents a circle centred at the origin with radius r. So the graph of the function: y=\sqrt{r^2-x^2} is therefore a semicircle.

Now if we rotate this around the x-axis through x=-r and x=r we will get a sphere. Therefore:

\displaystyle \text{V}=\pi\int^{\text{b}}_{\text{a}}y^2\text{d}x

\displaystyle =\pi\int^{\text{r}}_{\text{-r}}(\text{r}^2-x^2)\text{d}x

\displaystyle =\pi \begin{bmatrix}\text{r}^2x-\frac{x^3}{3}\end{bmatrix}^{\text{r}}_\text{-r}

\displaystyle =\frac{4\pi\text{r}^3}{3}

Which you should recognise as the formula for the volume of a sphere.

Volume of a Cone

Suppose we have a cone of radius r and height h.

Image:Function4.jpg

The equation of this line is going to be:

\displaystyle\text{y}=\frac{\text{r}x}{\text{h}}

The limits for our integration are going to be x=0 and x=h.

\displaystyle \text{V}=\pi\int^{\text{b}}_{\text{a}}y^2\text{d}x

\displaystyle =\pi\int^{\text{h}}_{\text{0}}\frac{(\text{r}x)^2}{\text{h}^2}\text{d}x

\displaystyle =\pi\int^{\text{h}}_{\text{0}}\frac{\text{r}^2x^2}{\text{h}^2}\text{d}x

\displaystyle =\pi \begin{bmatrix}\frac{\text{r}^2x^3}{3\text{h}^2}\end{bmatrix}^{\text{h}}_0

\displaystyle =\frac{\pi\text{r}^2\text{h}}{3}

Which is the equation for the volume of a cone, as expected.


Rotation about the y-axis

We have seen how to calculate a volume of revolution when a function is rotated about the x-axis, but what about if it is rotated about the y-axis?

In order to carry this out we must swap the roles of x and y.

Image:Function3.jpg

Before we can continue you must make sure two rules are met: -Firstly, the equation must be rearranged into the form: x=f(y) rather than y=f(x). -The limits must be given in terms of y, in this case y=c and y=d.

The formula for the volume then becomes:

\displaystyle \text{V}=\pi\int^{\text{d}}_{\text{c}}x^2\text{d}y


Examples

Will add later.

Comments

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