Calculus

Find the volume of the solid whose base is the circle x^2+y^2=64 and the cross sections perpendicular to the x-axis are triangles whose height and base are equal.
Find the area of the vertical cross section A at the level x=7.

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  1. each triangle at a distance x from (0,0) has a base of 2y = 2√(64-x^2)

    So, the total volume, taking advantage of symmetry is

    v = 2∫[0,8] bh/2 dx
    = 2∫[0,8] 2√(64-x^2)*2√(64-x^2)/2 dx
    = 4∫[0,8] (64-x^2) dx
    = 4096/3

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