Question

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

Answer 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.

Answer 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