Final answer:
The volume measurement using cylindrical slabs was smaller than that of a sphere due to the approximation method. More accurate techniques include applying calculus, utilizing geometric integration, employing spherical coordinates, and implementing numerical integration methods for a closer approximation.
Step-by-step explanation:
The volume of a fountain calculated using cylindrical slabs may have been smaller than the volume of an ideal sphere because a cylinder inscribed inside a sphere does not fill the entire space of the sphere. Hence, it provides only an approximation of the sphere's volume. A more accurate method for approximating the volume of a spherical slab would be:
- Applying advanced calculus techniques: Using calculus, we can integrate the volume element in spherical coordinates, offering a precise formula for the volume of a sphere.
- Utilizing geometric integration: By dividing the sphere into infinitesimally thin spherical shells and summing up their volumes, we can approximate the volume more closely than with cylindrical slabs.
- Employing spherical coordinates: Spherical coordinates naturally suit the geometry of a sphere, and using them can lead to an exact expression for volume.
- Implementing numerical integration methods: If analytical methods are too complex, numerical integration methods like the Monte Carlo simulation or Simpson's rule can be employed for approximation.