Volume of a Cylinder
Volume of a Cylinder
Using the definition of volume, we can calculate the sum of unit squares for any object with a closed space. In the case of a cylinder, a cylinder has a defined radius and height.
In the simulation below, you can input an arbitrary radius and height of your choice. To calculate the volume you can count each individual unit cubes to get the volume of a cylinder. Although we would get the correct answer, this is not practical as a cylinder with a large enough radius and height would take a long time to count up all the individual unit cubes each of length, width and height 1m by 1m by 1m.
Since a cylinder is a three dimensional representation of a circle, and to plot the radius of a circle reuires two dimensions and the height to form a rectangular figure a third dimension, we can assume that a cylinder will be in three dimensions. Since every cylinder has an radius r and a height h, we can take the integral of the height to find the volume of a cylinder.
Taking the integral of the area of a circle, we can then calculate the integral and find the volume of a cylinder:
\[ A = \pi r^2 \]
\[ V = \int_{0}^{h} \pi r^2 dx \]
\[ V = \pi r^2 \int_{0}^{h} dx \]
\[ V = \pi r^2 x_{0}^{h} \]
\[ V = \pi r^2 (h - 0) \]
\[ V = \pi r^2 h \]
An equation we can create therefore to calculate the volume of a cylinder is the radius squared times one third of the height times π.
\[ V = \pi r^2 h \]
Since a cylinder is an enclosed space in three dimensions, that would mean that the units would be meters cubed.
\[ m^3 \]
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