Amit Mallinath Navindgi , 16 Years ago
Grade 12
Analytical Geometry> how do we derive the formula for volume o...
2 Answers
Vijay Luxmi Askiitiansexpert
Last Activity: 16 Years ago
Dear Amit ,
Please post your query again , we are unable to understand .
AskiitianExpert Shine
Last Activity: 16 Years ago
Hi
VOLUME OF A SPHERE( its formula can be derived using integral calculus)
At any given x, the incremental volume (δV) is given by the product of the cross-sectional area of the disk at x and its thickness (δx):
The total volume is the summation of all incremental volumes:
In the limit as δx approaches zero this becomes:
At any given x, a right-angled triangle connects x, y and r to the origin, whence it follows from Pythagorean theorem that:
Thus, substituting y with a function of x gives:
This can now be evaluated:
![\!V = \pi \left[r^2x - \frac{x^3}{3} \right]_{x=0}^{x=r} = \pi \left(r^3 - \frac{r^3}{3} \right) = \frac{2}{3}\pi r^3.](http://upload.wikimedia.org/math/d/a/0/da0ec8ffc1583ad523f77280e451c6cf.png)
This volume as described is for a hemisphere. Doubling it gives the volume of a sphere as:
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