Revision:Volumes of Revolution

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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).

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).

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:

Examples

Volume of a Sphere

The equation: represents a circle centred at the origin with radius r. So the graph of the function: 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:

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.

The equation of this line is going to be:

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

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.

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:

Examples

Will add later.

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