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Math Help - Volumes of revolution

  1. #1
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    Volumes of revolution

    Hello all,
    New here but needing some help on a few questions in involving areas no in the first quadrant.
    Question 1: Find the volume of the solid generated by revolving the region bounded by y= -x^2-1 and y=x^2-2x-5 about the line y=11.

    Question 2: is using the same regions about x=4. The bounded region is in quadrants 3&4 and this seems to be throwing me off.
    Thanks in advance for your help.
    ninelegs
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  2. #2
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    Re: Volumes of revolution

    Quote Originally Posted by ninelegs View Post
    Hello all,
    New here but needing some help on a few questions in involving areas no in the first quadrant.
    Question 1: Find the volume of the solid generated by revolving the region bounded by y= -x^2-1 and y=x^2-2x-5 about the line y=11.

    Question 2: is using the same regions about x=4. The bounded region is in quadrants 3&4 and this seems to be throwing me off.
    Thanks in advance for your help.
    ninelegs
    start by making a sketch of the curves relative to the rotation axis

    find where the curves intersect ...

    -x^2-1 = x^2-2x-5

    x = 2 , x = -1

    method of washers ...

    R(x) = 11-(x^2-2x-5) = 16+2x-x^2

    r(x) = 11 - (-x^2-1) =  x^2+12

    V = \pi \int_{-1}^2 [R(x)]^2 - [r(x)]^2 \, dx


    about x = 4, use the method of cylindrical shells ...

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

    r(x) = 4-x

    h(x) = (-x^2-1)-(x^2-2x-5)
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  3. #3
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    Re: Volumes of revolution

    Thank you so much! I kept adding the distance from the axis of revolution and it was screwing me up.
    I have just one more for you.
    Find the volume of the solid generated by revolving the region bounded by the y-axis, the line y=1 and the curve x= y^2-4y about the line x=3.

    thanks again
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  4. #4
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    Re: Volumes of revolution

    Quote Originally Posted by ninelegs View Post
    Thank you so much! I kept adding the distance from the axis of revolution and it was screwing me up.
    I have just one more for you.
    Find the volume of the solid generated by revolving the region bounded by the y-axis, the line y=1 and the curve x= y^2-4y about the line x=3.

    thanks again
    you make a sketch and set this one up ... it will be w/r to y
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