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Math Help - volume problems!!

  1. #1
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    volume problems!!

    1.)Consider the given curves to do the following.
    Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.

    2.)Consider the given curves to do the following.
    Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about x = 4.

    3.)The region bounded by the given curves is rotated about the y-axis.
    Find the volume V of the resulting solid by any method.



    my main problem when doing these problems is i think im setting it up wrong at the beginning so can someone just help me set up these problems up to the point to the integral. Thanks in advance!
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  2. #2
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    Quote Originally Posted by ahawk1 View Post
    1.)Consider the given curves to do the following.
    Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.
    It will do the first one.

    The intersection between y = \tfrac{x^3}{125} and y=8 occurs when \tfrac{x^3}{125} = 8 \implies x^3 = 125\cdot 8 \implies x = 5\cdot 2 = 10.
    Therefore, the x values is going from 0\leq x\leq 10.
    Now we ought to know which curve is on top, pick a point say x=1. Notice that \tfrac{(1)^3}{125} < 8 therefore y=8 is on top of y=\tfrac{x^3}{125}.

    The volume is given by, \pi \int_0^{10} \left( 8\right)^2 - \left( \tfrac{x^3}{125} \right)^2 dx
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