y = square root 256- x squared, from 7 to 16 about x axis

Answer 1

V = Int (pi r^2 dx) where
r = y = 256-x^2

V = pi * Int(256-x^2 dx)[7,16]
= pi*(256x - 1/3 x^3)[7,16]
= pi*[(4096-1365.33)-(1792-114.33)]
= pi*(2730.67 - 1677.67)
= 1053pi

Check: Volume of spherical cap of height h on a sphere of radius r is given by

V = pi/6 * h *(3r^2 + h^2)
= pi/6 * 9 * (3*207 + 81)
= 1053pi