Volume of a Sphere
Volume of a Sphere
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 sphere, a sphere has a defined radius.
In the simulation below, you can input an arbitrary radius of your choice. To calculate the volume you can count each individual unit cubes to get the volume of a sphere. Although we would get the correct answer, this is not practical as a sphere with a large enough radius 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 sphere is a three dimensional representation of a circle, and to plot the radius and two angles require three dimensions, we can assume that a sphere will be in three dimensions. Since every sphere has an angle theta radians of π, an angle phi radians of 2π, and a radius r, we can take the integral of the radius to find the volume of a sphere.
We can first start out by writing the equation of a circle:
\[ x^2 + y^2 = r^2 \]
Solving for y we get:
\[ y^2 = r^2 - x^2 \]
\[ y = \sqrt{r^2 - x^2} \]
We can now take the integral of the area of a circle, and replacing r with y, we can then calculate the integral and find the volume of a sphere:
\[ A = \pi r^2 \]
\[ V = \int_{-r}^{r} \pi r^2 dx \]
\[ V = \int_{-r}^{r} \pi y^2 dx \]
\[ V = \int_{-r}^{r} \pi (\sqrt{r^2 - x^2})^2 dx \]
\[ V = \int_{-r}^{r} \pi (r^2 - x^2) dx \]
\[ V = \int_{-r}^{r} \pi r^2 - \pi x^2 dx \]
\[ V = \int_{-r}^{r} \pi r^2 dx - \int_{-r}^{r} \pi x^2 dx \]
\[ V = [\pi r^2 x]_{-r}^{r} - \left[\pi \frac{x^3}{3}\right]_{-r}^{r} \]
\[ V = [\pi r^3 - (-\pi r^3)] - \left[\pi \frac{r^3}{3} - \left(- \pi \frac{r^3}{3}\right)\right] \]
\[ V = 2\pi r^3 - \frac{2 \pi r^3}{3} \]
\[ V = \frac{6 \pi r^3}{3} - \frac{2 \pi r^3}{3} \]
\[ V = \frac{4 \pi r^3}{3} \]
\[ V = \frac{4}{3} \pi r^3 \]
An equation we can create therefore to calculate the volume of a sphere is the radius cubed times four thirds times π.
\[ V = \frac{4}{3} \pi r^3 \]
Since a sphere is an enclosed space in three dimensions, that would mean that the units would be meters cubed.
\[ m^3 \]
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