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Math Help - Volumes by the method of cross sections

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    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.
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    Quote Originally Posted by dalbir4444 View Post
    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 ...

    x^2 + y^2 = a^2

    using cross-sections perpendicular to the x-axis ...

    cross-section side length = 2y

    cross-sectional area = 4y^2

    cross-sectional thickness = dx

    V = \int_{-a}^a  4y^2 \, dx

    y^2 = a^2 - x^2

    V = 4\int_{-a}^a  a^2 - x^2 \, dx

    using symmetry ...

    V = 8\int_0^a  a^2 - x^2 \, dx

    integrate and evaluate using the FTC ... remember that a is a constant.
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