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Math Help - Volume of rotation (y-axis)

  1. #1
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    Volume of rotation (y-axis)

    The function is y(x) = 7/(x^2 +5x +6) evaluated from x = 0 to x = 1. Obviously it cannot be written as x(y). I tried changing the bounds of integration (from x values to their corresponding y values on the function), but this didn't work.

    How do you do this? Please explain this in steps. Thanks!
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  2. #2
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    Quote Originally Posted by iheartmath View Post
    The function is y(x) = 7/(x^2 +5x +6) evaluated from x = 0 to x = 1. Obviously it cannot be written as x(y). I tried changing the bounds of integration (from x values to their corresponding y values on the function), but this didn't work.

    How do you do this? Please explain this in steps. Thanks!
    You can split the denominator into partial fractions using

    x^2+5x+6=(x+3)(x+2)

    \frac{1}{(x+3)(x+2)}=\frac{A}{x+3}+\frac{B}{x+2}

    which gives

    A+B=0
    2A+3B=1

    Simply solve for A and B.

    Then you can develop the volume of revolution using shells.
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  3. #3
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    Quote Originally Posted by iheartmath View Post
    The function is y(x) = 7/(x^2 +5x +6) evaluated from x = 0 to x = 1. Obviously it cannot be written as x(y). I tried changing the bounds of integration (from x values to their corresponding y values on the function), but this didn't work.

    How do you do this? Please explain this in steps. Thanks!
    rotating about the y-axis, using the method of cylindrical shells ...

    V = 2\pi \int_0^2 \frac{7x}{x^2+5x+6} \, dx

    partial fractions ...

    \frac{7x}{(x+3)(x+2)} = \frac{21}{x+3} - \frac{14}{x+2}
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