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

$$Howtoderivetheformula$$$$forfindingthevolumeof$$$$sphericalcurl?$$

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

$$inqn125322$$$$V=\pi h^2(R-\frac{h}{3}).howgotit?$$

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

$$\frac{x}{r}=\frac{r}{2R-x}$$$$\Rightarrow r^2=x(2R-x)$$$$dV=\pi r^{2}dx$$$$V=\pi {\int }_0^h{r}^{2}dx=\pi {\int }_0^{h}x(2R-x)dx$$$$=\pi {[{Rx}^2-\frac{x^3}{3}]}_0^h$$$$=\pi ({Rh}^2-\frac{h^3}{3})$$$$=\pi h^2(R-\frac{h}{3})\Rightarrow proved$$$$$$$$withh=R:$$$$V=\pi R^2(R-\frac{R}{3})=\frac{2\pi R^3}{3}=hemisphere$$$$withh=2R:$$$$V=\pi 4R^2(R-\frac{2R}{3})=\frac{4\pi R^3}{3}=sphere$$

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

Commented by bemath last updated on 10/Dec/20

$$waw....thankssir$$

Commented by bramlexs22 last updated on 11/Dec/20

$$inotherway$$$$V=\pi \underset{R-h}{\stackrel{R}{\int }}(R^2-x^2)dx$$$$\text{Missing left or extra right}$$$$=\pi x{\left(\frac{3R^2-x^2}{3}\right)}_{R-h}^R$$$$=\pi [\frac{2R^3}{3}-(R-h)\left(\frac{3R^2-R^2+2Rh-h^2}{3}\right)]$$$$=\frac{2\pi R^3}{3}-\frac{1}{3}\pi (R-h)(2R^2+2Rh-h^2)$$$$=\frac{2\pi R^3}{3}-\frac{1}{3}\pi (2R^3-3{Rh}^2+h^3)$$$$=\pi {Rh}^2-\frac{1}{3}\pi r^3$$$$=\pi h^2(R-\frac{h}{3}).$$

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