When you take a fraction such as y = f(x) or between two limits on the x-axis and rotate this part of the curve around the x-axis the shape formed will be a 3 dimensional shape. In this lesson I shall be showing how to find the volumes for solids of revolution about the axis on the graph such as the x-axis or the y-axis.

For example

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As we can see from the above solids on the graph the 3D object has a cross-section of a circle at every x-axis point. The concept of finding the volumes works very similar to finding volumes using the normal formulas.
You must know that we can find areas by integration; well we can also find volumes by integration. The general formula for finding volume of revolution for y = f(x) is
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Notice we’re just integrating as we did with finding the area below the curve and any axis.

Example

Find the volume of the solid formed when the function y = x2 is rotated around the x-axis between x=1 and x=3.
The solid we’re trying to find the volume for is shown on the graph below.
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We know that the volume is given by:
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The question says y=x2 which means the volume will become;
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Now we integrate;
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Notice we have taken π out before the integration.
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It’s that easy!

Example 2

y=ex between x = 0 and x=2
This function forms the shape below;
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We know;
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It’s best to leave the answers in a simplified form unless told otherwise.

Revolution about the y-axis

You can also find volumes for curves rotated around the y-axis. The following diagram shows these kind of shapes for the functions of y=f(x).
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Remember the curve has been rotated around the y-axis instead of the x-axis. So this will have to affect the formula to find the volume as well.
If we’re rotating about the y-axis the formula becomes.
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Example

Find the volume of the solid formed when the curve y-/x is rotated about the y-axis between x=0 and x=9.
In this case we also have to find the limits for the y-axis and use, instead of the given x-axis limits which have been given.
Here is the curve;
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Now let’s find the y limits; when x = 0, y = /9 = 0 when x = 9, y = /9 = 3.
This will be the solid formed from the rotation about the y-axis.
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Volume can be given by;
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If y = /x then x = y2
Our integral becomes;
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