calculus

the base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line y=1-x. if cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?

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  1. area of semicircle = (1/2) pi r^2 = (1/2) pi D^2/4 = (pi/8) D^2
    I used D instead of r because the solid seems to be made up of circles sitting on this triangle with radius = y/2 and centered at y/2
    So cross sectional area =(pi/8)y^2
    so dV = (pi/8) dx [ (1-x)^2 ]
    integrate from x = 0 to x = 1
    (pi/8) [ 1 dx - 2 x dx + x^2 dx ]
    =(pi/8) [ x -x^2 + (1/3)x^3 ] at x = 1 because all terms are 0 at x = 0
    = (pi/8)(1/3)
    = pi/24

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    posted by Damon
  2. Integrate pi(1-x)^2/2 dx from 0 to 1.
    (let u = 1-x )
    V = Integral pi u^2/2 du from 0 to 1
    = pi/6

    Each pi (1-x)^2 dx slab is a slab perpendicualr to the x axis

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    posted by drwls
  3. Damon is right. The individual slab area are pi (1-x)^2/8. I confused diameter with radius

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    posted by drwls

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