In the syllabus of A-levels' Further Mathematics course, there is a chapter that includes the calculation of the coordinates of centroids of two-dimensional graphs. Albeit this operation looks simple at a glance, it may not make much sense at hindsight. There is a way of getting around all the complicated thinking if you are introduced to the simple meaning of a centroid. Geometrically, when a graph is rotated over an axis, the volume of the solid of revolution formed is the product of the area underneath the curve and the distance that the centroid travels. Cutting this into simpler bits, Volume of revolution = Area underneath graph x distance traveled by centroid.
Firstly, what is the volume of revolution of a graph about an axis? Well, most A2 level Mathematics students should already know it by now, as it is covered near the end of the AS-level Mathematics course. For those who do not know, it is the volume of the solid obtained when a two-dimensional graph, limited by two lines portruding from points on the axis (vertical if from the x-axis, horizontal if from the y-axis). Although it may sound complicated, there is a quite simple formula and a very comprehensible derivation. The aforementioned can be found in most elementary calculus textbooks and websites so I will not go into much detail on the derivation. The formula, however, is as follows:
Second, we have to identify the area covered by the graph within said limits. Relative to the Volume of Revolution, finding out the Area under a graph is a much simpler task. The area under a graph is simply the integration, with respect to x, of the function (in terms of x) of the graph within the required limits. In other words:
So, recalling our previous relationship, V = A x distance traveled by the centroid.
Now, what is the distance traveled by the centroid?
If you can recall correctly, it is a point on the graph itself. So, when the graph rotates, so does the centroid! As a matter of fact, when a point revolves across an axis, it forms a circle! So, the distance traveled by the centroid is actually the circumference of the circle it forms! Now, lets try to visualize this for a second. If you cannot visualize it clearly, I'll provide a visual aid:
The circle at the center of the graph is the centroid's image when it revolves around the x-axis. On the graph, the limits are set as a and b and the centroid is the white dot that's a distance r from the x-axis. Now, treating the centroid as a point on the graph, we know that its y coordinate is r. Now, following the previous formula V = A x 2πr, we can find r once we know V and A. Hence, once we have the limits and the equation of the curve, finding the centroid is just a matter of solving the aforementioned equation.
Conclusively:
This, however, only gives the y-coordinate of the centroid. I trust that after reading this, you will be able to find the x-coordinate using a similar operation because Pappus' theorem applies to all cases involving centroids.
Please leave questions and comments; I'll be more than glad to address them!
Firstly, what is the volume of revolution of a graph about an axis? Well, most A2 level Mathematics students should already know it by now, as it is covered near the end of the AS-level Mathematics course. For those who do not know, it is the volume of the solid obtained when a two-dimensional graph, limited by two lines portruding from points on the axis (vertical if from the x-axis, horizontal if from the y-axis). Although it may sound complicated, there is a quite simple formula and a very comprehensible derivation. The aforementioned can be found in most elementary calculus textbooks and websites so I will not go into much detail on the derivation. The formula, however, is as follows:
Second, we have to identify the area covered by the graph within said limits. Relative to the Volume of Revolution, finding out the Area under a graph is a much simpler task. The area under a graph is simply the integration, with respect to x, of the function (in terms of x) of the graph within the required limits. In other words:
So, recalling our previous relationship, V = A x distance traveled by the centroid.
Now, what is the distance traveled by the centroid?
If you can recall correctly, it is a point on the graph itself. So, when the graph rotates, so does the centroid! As a matter of fact, when a point revolves across an axis, it forms a circle! So, the distance traveled by the centroid is actually the circumference of the circle it forms! Now, lets try to visualize this for a second. If you cannot visualize it clearly, I'll provide a visual aid:
The circle at the center of the graph is the centroid's image when it revolves around the x-axis. On the graph, the limits are set as a and b and the centroid is the white dot that's a distance r from the x-axis. Now, treating the centroid as a point on the graph, we know that its y coordinate is r. Now, following the previous formula V = A x 2πr, we can find r once we know V and A. Hence, once we have the limits and the equation of the curve, finding the centroid is just a matter of solving the aforementioned equation.
Conclusively:
This, however, only gives the y-coordinate of the centroid. I trust that after reading this, you will be able to find the x-coordinate using a similar operation because Pappus' theorem applies to all cases involving centroids.
Please leave questions and comments; I'll be more than glad to address them!
