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28 March, 11:20

The base of S is the triangular region with vertices (0, 0), (1, 0), and (0, 1). Cross-sections perpendicular to the x-axis are squares. Find the volume V of this solid.

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  1. 28 March, 14:18
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    the volume of the solid is V=1/6

    Step-by-step explanation:

    The solid S has a triangular cross section in the xy-plane with sides of length L=1. The boundaries are

    x=0

    y=0

    y = 1-x

    Since each cross section perpendicular to the x axis (parallel to the yz-plane) is a square then:

    z=x

    then the volume of the solid will be

    V = ∫dV=∫∫∫dxdydz ∫₀¹ (∫₀¹⁻ˣ dy) (∫₀ˣdz) dx = ∫₀¹ (1-x) * x dx = ∫₀¹ (x-x²) dx = [ (1/2) x² - (1/3) x³] |₀¹ = 1/2 - 1/3 = 1/6

    V=1/6
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