1. Deriving the Sphere Volume Formula via Calculus
Consider a sphere centered at the origin of radius \(r\). A horizontal slice through the sphere at height \(x\) is a circular disk with radius \(\sqrt{r^2 - x^2}\). The area of this disk is:
A(x) = π (√(r² - x²))² = π(r² - x²)
Integrating these infinitesimal circular disks from \(x = -r\) to \(x = +r\) yields the total volume:
V = \int_{-r}^{r} \pi(r^2 - x^2) \, dx = 2\pi \int_{0}^{r} (r^2 - x^2) \, dx
\( V = 2\pi \left[ r^2 x - \frac{x^3}{3} \right]_0^r = 2\pi \left( r^3 - \frac{r^3}{3} \right) = 2\pi \left( \frac{2}{3}r^3 \right) = \mathbf{\frac{4}{3}\pi r^3} \).
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