# Thread: Volumes by the method of cross sections

1. ## Volumes by the method of cross sections

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.

2. Originally Posted by dalbir4444
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.