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Math Help - Volume of solid generated by curves. ( I tried my best!)

  1. #1
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    Volume of solid generated by curves. ( I tried my best!)

    Q. The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid.

    y=x^3 and y=sqrt.of(x); about x = 1.

    What I did: Rotating about a vertical line along the y-axis.
    x^3=sqrt.of(x); x= 0 and 1
    x=y^(1/3) and x=y^2.
    A(y)=pi(outer radius)^2-pi(inner radius)^2
    =pi(1+y^(1/3))^2-pi(1+y^2)^2
    V= pi*Integral from 0 to 1 of [(1+(y^1/3))^2-(1+y^2)^2]dy
    V= pi*Integral from 0 to 1 of [(1+y^(2/3)+2y^(1/3)-(1+y^4+2y^2)]dy
    V= pi[(3y^(2/3))/2 + (6y^(4/3))/4 - (y^4)/4 - (2y^3)/3] from 0 to 1

    But after plugging in for 0 and 1, I got 25pi/12. But the ans. is 17pi/30...8(
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  2. #2
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    but that's not the answer though. Plus are you able to do it my way?
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  3. #3
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    I wonder why you put 1+y^{1/3}... But since I never worked with solid or rotation other than about the main axes, I would have gone with this:

    From what I see, you need to move both curves to the left by one unit.

    Hence, the equation of the new cubic becomes:  y = (x+1)^3

    And that of the square root becomes: y = \sqrt{x+1}

    The volume then becomes:

    \displaystyle \pi \int^1_0 (1-y^2)^2 - (1-y^{\frac13})^2\ dy

    Which gives:

    \displaystyle \pi \int^1_0 y^4-2y^2  - y^{\frac23}+2y^{\frac13}\ dy

    \displaystyle \pi \left[\dfrac15y^5 - \dfrac23y^3 - \dfrac35y^{\frac53}+\dfrac32y^{\frac43}\right]^1_0

    \displaystyle \pi \left[\left(\dfrac15 - \dfrac23 - \dfrac35+\dfrac32 \right) -(0) \right]^1_0

    Are you sure the answer is 17pi/30? I'm getting 13pi/30...

    Or I missed something...
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  4. #4
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    method of cylindrical shells w/r to x ...

    \displaystyle V = 2\pi \int_0^1 (1-x)(\sqrt{x} - x^3) \, dx = \frac{13\pi}{30}

    method of washers w/r to y ...

    \displaystyle V = \pi \int_0^1 (1 - y^2)^2 - (1 - \sqrt[3]{y})^2 = \frac{13\pi}{30}
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  5. #5
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    you guys are right. It's 13pi/30. Thanks a lot. Some dumb mistakes huh?
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