Answer to Question #155200 in Calculus for Phyroe

Question

Considering the volume of a spherical shell as an increment of volume of sphere, find approximately the volume of a spherical shell whose outer diameter is 8 inches and whose thickness is 1/16 inch.

Answer 1

Outer sphere volume: (4/3)*$\pi$ *R³, R - outer diameter.

Inner sphere volume: (4/3)*$\pi$ *r³, r - inner diameter.

The difference between the volumes will give the shell volume:

4/3*$\pi$ *(R³-r³) = 4/3*$\pi$*(R-r)*(R²+R*r + r²) -> (R-r) = 1/16 inch -> 4/3*$\pi$*(1/16)*(R²+R*r+r²) =

= (1/3*1/4)*$\pi$ *(8² + 8*$(8-\frac{1}{16})$ + (8-$\frac{1}{16} ​$ )² ) = 1/12 * (8² + 8² - $\frac{8}{16} ​$ + (8² - 2*8*$\frac{1}{16} ​$ + $\frac{1}{256} ​$))*$\pi$ =

= $\frac{1}{12} ​$ * (3*8² - 1.5 + $\frac{1}{256} ​$)*$\pi$ = ($\frac{1}{4} ​$*64 - 1.5 +$\frac{1}{3072} ​$) * $\pi$ = (14.5 + 0.0003) * $\pi$ $\approx$ 14.5003 * 3.14159 $\approx$ 45.554 cubic inches (inch³).

Answer: 45.554 inch³.