Math repost for Reina

This question was originally posted on another student's thread.

The cup of a wine glass has a shape formed by rotating the parabola y=x^2 about the y-axis. Its upper rim is a circle of radius 1 unit . How much wine can it hold ?

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posted by Reina
today at 3:09pm

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Ms. Sue
  1. clearly, the volume of the cup is found by rotating the curve
    y = x^2
    about the y-axis. Since the radius at the top is 1, the curve is from (0,0) to (1,1)
    Using discs of thickness dy, we have
    v = ∫[0,1] πr^2 dy
    where r = x
    v = ∫[0,1] πy dy = π/2
    Makes sense, since a cone of that radius would have volume π/3. and the goblet, being round instead of pointed, would hold more.

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  2. picture the cup as a stack of discs

    the thickness of a disc is dy ... the radius of a disc is x
    ... the volume of a disc is ... π x^2 dy
    ... y = x^2 ... dv = π y dy

    integrating from y = 0 to y = 1 ... v = π y^2 / 2

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