The volume of a solid is one of the most fundamental properties of any shape. Volumes can be expressed in cubic units, such as meters3 (m3) or cubic feet (ft3).
Knowing the volume of a shape allows you to calculate how much material it contains. For example, if you know the volume of a sphere, you can calculate how much water it contains. If you know the volume of a doughnut shape, you can calculate how big it is on top of your coffee.
Calculating the volume of complex shapes can be challenging. There are many cases where you need to know the specific orientation of parts of the shape with respect to the coordinate planes and each other in order to find the volume.
This article will discuss how to find the volume of a solid generated by revolving a region about an axis. We will first discuss what this means and give some examples, then we will go into detail on how to find the volume using algebraic methods.
Divide the shaded region into thin vertical strips
The next step is to divide the region into thin vertical strips. You will need to do this in every case, because you cannot find the volume of a solid generated by revolving a region about an axis if the region extends beyond the plane of the axis.
For example, imagine that the region being rotated about the X-axis extended down to where the Y-axis is. You could not find the volume of this solid because part of it would be outside of the solid being investigated.
In this case, we will have a total of n strips, where n is the number of horizontal lines in the original rectangle. We will call each strip a layer. The layers will go from top to bottom, with each one just covering one line of the original rectangle.
We need to make sure that each layer just covers one line of the rectangle or else it would not be able to be rotated about the X-axis.
Find the volume of one vertical strip
Now that you have the average thickness, you can find the volume of one vertical strip. You do this by dividing the total volume by the number of slices.
Imagine a tower of pizza slices: The height of the tower is equal to the sum of all the slices’ heights, and to demolish it, you have to break all those slices. Same thing here!
The volume of one vertical strip is just how much volume lies between one side of the cylinder and one slice of it. How do you know? Because when you cut through a cylinder, whether with a saw or a scalpel, what you get on either side is just more cylinder.
There may be some leftover bits that need to be trimmed off, but in general, breaking up a cylinder into thinner cylinders doesn’t change what it is: A cylinder.
Use calculus to find the volume of the entire solid
Once you have found the volume of the solid generated by revolving a region about an axis, you can use calculus to find the volume of the entire solid. Calculus is a field of math that deals with rates of change and functions.
Calculus is most often used in science, engineering, and medicine. Many college level courses require knowledge of calculus as well. It is a very useful field of math!
The trick in calculus to finding the volume of a solid generated by revolving a region about an axis is to find the average rate of change of the area of the region as it rotates. Then, you take some kind of integral (another kind of mathematics) with respect to time to find the total volume.
Check your work by calculating the volume using both formulas
Once you have the individual volumes, you can assemble them into the total volume of the solid. To do this, you must first make sure that your slices are equivalent – that is, that their heights and widths are proportional.
Then, you must determine the volume of one slice and how to add all of the slices’ volumes together to get the total volume.
You can do this by using either the cylinder or prism formula, depending on what is easier for you. Check your work by calculating the volume using both formulas and making sure they are equal!
Some students find it helpful to draw out both shapes on paper and move pieces around to see if the shape changes. This can help you see how the shapes break down into smaller pieces and where they match up.
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