How-to-derive-the-formula-for-finding-the-volume-of-spherical-curl-

Question Number 125368 by bemath last updated on 10/Dec/20

H o w t o d e r i v e t h e f o r m u l a

f o r f i n d i n g t h e v o l u m e o f

s p h e r i c a l c u r l ?

Commented by Ar Brandon last updated on 10/Dec/20

https://www.therightgate.com/deriving-curl-in-cylindrical-and-spherical/

Commented by bemath last updated on 10/Dec/20

Commented by bemath last updated on 10/Dec/20

i n q n 125322

V = π h^{2} (R - \frac{h}{3}) . h o w g o t i t ?

Commented by mr W last updated on 10/Dec/20

\frac{x}{r} = \frac{r}{2 R - x}

\Rightarrow r^{2} = x (2 R - x)

d V = π r^{2} d x

V = π \int_{0}^{h} r^{2} d x = π \int_{0}^{h} x (2 R - x) d x

= π {[{R x}^{2} - \frac{x^{3}}{3}]}_{0}^{h}

= π ({R h}^{2} - \frac{h^{3}}{3})

= π h^{2} (R - \frac{h}{3}) \Rightarrow p r o v e d

w i t h h = R :

V = π R^{2} (R - \frac{R}{3}) = \frac{2 π R^{3}}{3} = h e m i s p h e r e

w i t h h = 2 R :

V = π 4 R^{2} (R - \frac{2 R}{3}) = \frac{4 π R^{3}}{3} = s p h e r e

Commented by mr W last updated on 10/Dec/20

Commented by bemath last updated on 10/Dec/20

w a w \dots . t h a n k s s i r

Commented by bramlexs22 last updated on 11/Dec/20

i n o t h e r w a y

V = π \underset{R - h}{\int^{R}} (R^{2} - x^{2}) d x

Missing \left or extra \right

= π x {(\frac{3 R^{2} - x^{2}}{3})}_{R - h}^{R}

= π [\frac{2 R^{3}}{3} - (R - h) (\frac{3 R^{2} - R^{2} + 2 R h - h^{2}}{3})]

= \frac{2 π R^{3}}{3} - \frac{1}{3} π (R - h) (2 R^{2} + 2 R h - h^{2})

= \frac{2 π R^{3}}{3} - \frac{1}{3} π (2 R^{3} - 3 {R h}^{2} + h^{3})

= π {R h}^{2} - \frac{1}{3} π r^{3}

= π h^{2} (R - \frac{h}{3}) .