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Calculus Voulme

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find the volume of the solid generated by revolving the region r bounded by the graphs of the given equations about the y-axis.
x^2+y^2=1
x=1
y=1

  • Calculus Voulme -

    I see a cylinder of radius 1 and height 1, containing a half sphere.

    so volume of your generated solid
    = π(1^1)(1) - (1/2)(4/3)π(1^3)
    = π - (2/3)π
    = π/3

  • Calculus Voulme -

    same question by Calculus

    using "washers"
    outer radius = r1 = 1 --->r1^2 = 1
    inner radius = r2 = √(1-y^2) --->r2^2 = 1-y^2

    Volume = π∫(1 - (1-y^2) dy from y = 0 to 1
    = π∫y^2 dy from y = 0 to 1
    = π[ y^3/3] from 0 to 1
    = π(1/3 - 0) = π/3 , same as above

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