An observatory is shaped like a solid whose base is a circular disk with diameter AB of length 2a. Find the volume of the solid if each cross section perpendicular to AB is a square.
base equation ...
$\displaystyle x^2 + y^2 = a^2$
using cross-sections perpendicular to the x-axis ...
cross-section side length = $\displaystyle 2y$
cross-sectional area = $\displaystyle 4y^2$
cross-sectional thickness = $\displaystyle dx$
$\displaystyle V = \int_{-a}^a 4y^2 \, dx$
$\displaystyle y^2 = a^2 - x^2$
$\displaystyle V = 4\int_{-a}^a a^2 - x^2 \, dx$
using symmetry ...
$\displaystyle V = 8\int_0^a a^2 - x^2 \, dx$
integrate and evaluate using the FTC ... remember that $\displaystyle a$ is a constant.