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Calculating volume

Posted on | June 1, 2009 |

do the math: it's big

do the math: it's big

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically.
The volume of a sphere is the integral of infinitesimal circular slabs of thickness. The calculation for the volume of a sphere with center 0 and radius r is as follows. The radius of the circular slabs is y = sqrt{r^2-x^2} The surface area of the circular slab is ?y2. The volume of the sphere can be calculated as int_{-r}^r pi(r^2-x^2) ,dx = int_{-r}^r pi r^2,dx - int_{-r}^r pi x^2 ,dx

Now int_{-r}^r pi r^2,dx = 2pi r^3
and int_{-r}^r pi x^2 ,dx = 2 pi frac{r^3}{3}

Combining yields left(2-frac{2}{3}right)pi r^3 = frac{4}{3}pi r^3 This formula can be derived more quickly using the formula for the sphere’s surface area, which is 4?r2. The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to int_0^r 4pi r^2 ,dr = frac{4}{3}pi r^3

Category: Bigassology
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