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Thread: Rotational solids? Drawing a blank...

  1. #1
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    Rotational solids? Drawing a blank...

    Just needed to check something with you math experts ;o)

    1. Find the volume of the solid obtained by rotating the region bounded by $\displaystyle y=\sqrt[6]{x}$ and $\displaystyle y=x$, rotated about the line $\displaystyle y=1$

    Is it

    $\displaystyle \int_0^1\pi{(1-\sqrt[6]{x})-(1-x)}$ ??

    2. Same premise as the previous problem, but with the following data:
    region bounded by $\displaystyle y=x^4$ and $\displaystyle x=y^4$, rotated about $\displaystyle x=-1$

    Now, this really got me. Am I supposed to put it all in terms of $\displaystyle x$, so that the regions would be $\displaystyle y=x^4$ and $\displaystyle y=\sqrt[4]{x}$ ?? I'm having a little trouble graphing these... but I think the lower limit is zero...?

    Lastly,
    3. When I'm told to use the method of cylindrical shells to find a volume, is that the following equation:

    $\displaystyle \int_a^b{2\pi}({r})({h})dx$ ??

    Thank you for your help!
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  2. #2
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    Quote Originally Posted by Kimberly View Post
    Just needed to check something with you math experts ;o)

    1. Find the volume of the solid obtained by rotating the region bounded by $\displaystyle y=\sqrt[6]{x}$ and $\displaystyle y=x$, rotated about the line $\displaystyle y=1$

    Is it

    $\displaystyle \int_0^1\pi{(1-\sqrt[6]{x})-(1-x)}$ ??

    2. Same premise as the previous problem, but with the following data:
    region bounded by $\displaystyle y=x^4$ and $\displaystyle x=y^4$, rotated about $\displaystyle x=-1$

    Now, this really got me. Am I supposed to put it all in terms of $\displaystyle x$, so that the regions would be $\displaystyle y=x^4$ and $\displaystyle y=\sqrt[4]{x}$ ?? I'm having a little trouble graphing these... but I think the lower limit is zero...?

    Lastly,
    3. When I'm told to use the method of cylindrical shells to find a volume, is that the following equation:

    $\displaystyle \int_a^b{2\pi}({r})({h})dx$ ??

    Thank you for your help!

    #1 ...

    disks w/r to x

    $\displaystyle \displaystyle V = \pi \int_0^1 (1-x)^2 - (1-\sqrt[6]{x})^2 \, dx
    $

    shells w/r to y

    $\displaystyle \displaystyle V = 2\pi \int_0^1 (1-y)(y-y^6) \, dy$


    #2 might be easier using shells w/r to x

    $\displaystyle \displaystyle V = 2\pi \int_0^1 (x+1)(\sqrt[4]{x} - x^4) \, dx$


    #3 $\displaystyle \displaystyle V = 2\pi \int_a^b r(x) \cdot h(x) \, dx$
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