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Thread: Help with finding volumes of rotated curves

  1. #1
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    Help with finding volumes of rotated curves

    Hey I was recently introduced to the disc, cylindrical shell, and washer methods and I don't understand them fully mostly because it's difficult for me to imagine the shapes created when rotating curves and i dont know which method to use. Here is the problem:

    Consider the solid obtained by rotating the region bounded by the given curves about the x-axis.
    Find the volume V of this solid.

    I decided to use the disc method and set up my integral like so:

    $\displaystyle \pi\int(16-4x^2)^2dx$

    With -2 and 2 for the limits of integration...solving this yields 64$\displaystyle \pi$ but i cant get the correct volume.
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  2. #2
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    1) I always try to do it both ways, just to emphasize the concepts in my mind.

    2) That is not $\displaystyle 64\pi$. Please demonstrate your work.

    3) You have it set up exactly correctly. Try the integral again.

    4) And the other way: $\displaystyle 2\pi \int_{0}^{16}y \left[ 2\sqrt{\frac{16-y}{4}}\right] \;dy$. See if you get the same answer.

    5) Please learn to observe an exploit symmetries. The limits of your integral could have been [0,2] with the result multiplied by 2.
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  3. #3
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    Thank you for the helpful reply i see what i did, or rather, didn't do.
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