Investigation # 2 – Volume of Solids of Rotation – Prachi Dave
INTRODUCTION
The volume of a solid of rotation is when the region of an object is rotated about a line, in the
same plane. For example, a solid right circular cylinder can be generated by revolving a rectangle.
Similarly, a solid spherical ball can be generated by revolving a semi-disk.
There are two ways of calculating this volume: the Disk Method and the Washer Method. The Disk
Method is used when there is a single function (i.e., f(x)) revolving around an axis, whereas the
Washer Method is when there are two functions (i.e., f(x) and g(x)) revolving around an axis.
To find the volume of any solid, using the Disk Method, the formula is
In this investigation, the task is to calculate the volume of a water
carafe using different methods and comparing the results obtained
from both ways.
As the capacity of this carafe is 1.33 L, the volume should be to close
to 0.00133 m3 or 1330 cm3.
For this solid, the Disk method will be used as there is only one
function available. For this investigation, to find the volume of the
carafe, GeoGebra and the CAS calculator will be used, as well using
the Trapezium Rule to approximate the volume.
METHOD ONE – GEOGEBRA
The first step is to place the image next to the y-axis and to make sure the end of the carafe is at
the origin. For this investigation, the solid is revolving around the x-axis, so the object has to be
placed on the x-axis, with half of it in the first quadrant and the other in the fourth quadrant. Only
the positive side’s “curve” will be used to calculate the function.
Use place points to create a curve on only one side of the axis and then delete the image of the
carafe. It should now look like what can be seen below.
, Next open a spreadsheet and enter all the coordinate points. Then select the points and graph.
Set the regression model to ‘polynomial’ and the degree of the polynomial to ‘4’, as that gives the
best curve for the carafe.
The function of this graph is also given:
Using the formula , the volume can be calculated over the interval x E [0, 22.0].
Calculate the volume using a CAS or similar. For this graph, the volume is 1020.93 cm 3.
∴ The volume is 0.001021 m3.
This answer is accurate as it is close to the volume, when it is converted by the capacity.
However, it is not the closest possible. So this needs to be done again, but this time, only using
the turning points of the carafe.
The points that will be used are (0, 2.6), (9, 6.1), (18, 3.2) and (21.2).
INTRODUCTION
The volume of a solid of rotation is when the region of an object is rotated about a line, in the
same plane. For example, a solid right circular cylinder can be generated by revolving a rectangle.
Similarly, a solid spherical ball can be generated by revolving a semi-disk.
There are two ways of calculating this volume: the Disk Method and the Washer Method. The Disk
Method is used when there is a single function (i.e., f(x)) revolving around an axis, whereas the
Washer Method is when there are two functions (i.e., f(x) and g(x)) revolving around an axis.
To find the volume of any solid, using the Disk Method, the formula is
In this investigation, the task is to calculate the volume of a water
carafe using different methods and comparing the results obtained
from both ways.
As the capacity of this carafe is 1.33 L, the volume should be to close
to 0.00133 m3 or 1330 cm3.
For this solid, the Disk method will be used as there is only one
function available. For this investigation, to find the volume of the
carafe, GeoGebra and the CAS calculator will be used, as well using
the Trapezium Rule to approximate the volume.
METHOD ONE – GEOGEBRA
The first step is to place the image next to the y-axis and to make sure the end of the carafe is at
the origin. For this investigation, the solid is revolving around the x-axis, so the object has to be
placed on the x-axis, with half of it in the first quadrant and the other in the fourth quadrant. Only
the positive side’s “curve” will be used to calculate the function.
Use place points to create a curve on only one side of the axis and then delete the image of the
carafe. It should now look like what can be seen below.
, Next open a spreadsheet and enter all the coordinate points. Then select the points and graph.
Set the regression model to ‘polynomial’ and the degree of the polynomial to ‘4’, as that gives the
best curve for the carafe.
The function of this graph is also given:
Using the formula , the volume can be calculated over the interval x E [0, 22.0].
Calculate the volume using a CAS or similar. For this graph, the volume is 1020.93 cm 3.
∴ The volume is 0.001021 m3.
This answer is accurate as it is close to the volume, when it is converted by the capacity.
However, it is not the closest possible. So this needs to be done again, but this time, only using
the turning points of the carafe.
The points that will be used are (0, 2.6), (9, 6.1), (18, 3.2) and (21.2).
