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Math Help - Need help setting up a shells problem

  1. #1
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    Need help setting up a shells problem




    EDIT: Please excuse my typo, I meant integrate from 1 to 5.
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  2. #2
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    Since you are rotating around the \displaystyle y axis, this will be a \displaystyle dy integral.

    The graphs intersect where the equations are equal...

    \displaystyle x = 1 + (x-3)^2

    \displaystyle x = 1 + x^2 - 6x + 9

    \displaystyle 0 = x^2 - 7x + 10

    \displaystyle 0 = (x - 2)(x - 5)

    \displaystyle x = 2 or \displaystyle x = 5.

    When \displaystyle x = 5, y = 2, so they intersect at \displaystyle (x,y) = (5, 2).


    The volume is calculated by rotating the area around the \displaystyle y axis.

    This area needs to be thought of as a series of rectangles, which when rotated, become cylinders.

    The length of each rectangle is \displaystyle (y + 3) - (1 + y^2) = 2 + y - y^2 and the width is \displaystyle dy, a small change in \displaystyle y.

    When rotated, each cylinder has a radius the same as the length of each rectangle, so the circular cross-sectional area is \displaystyle \pi r^2 = \pi (2 + y - y^2)^2. The height of each cylinder is the same as the width of each rectangle, \displaystyle dy.

    Therefore the volume of each cylinder is \displaystyle \pi (2 + y - y^2)^2\,dy.

    So the entire volume can be approximated by \displaystyle \sum{\pi(2+y-y^2)^2\,dy}, and when you increase the number of cylinders, this sum converges on an integral and the approximation becomes exact.

    So \displaystyle V = \int_0^2{\pi (2 + y - y^2)^2\,dy}.
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  3. #3
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    Quote Originally Posted by Prove It View Post
    Since you are rotating around the \displaystyle y axis, this will be a \displaystyle dy integral.

    The graphs intersect where the equations are equal...

    \displaystyle x = 1 + (x-3)^2

    \displaystyle x = 1 + x^2 - 6x + 9

    \displaystyle 0 = x^2 - 7x + 10

    \displaystyle 0 = (x - 2)(x - 5)

    \displaystyle x = 2 or \displaystyle x = 5.

    When \displaystyle x = 5, y = 2, so they intersect at \displaystyle (x,y) = (5, 2).


    The volume is calculated by rotating the area around the \displaystyle y axis.

    This area needs to be thought of as a series of rectangles, which when rotated, become cylinders.

    The length of each rectangle is \displaystyle (y + 3) - (1 + y^2) = 2 + y - y^2 and the width is \displaystyle dy, a small change in \displaystyle y.

    When rotated, each cylinder has a radius the same as the length of each rectangle, so the circular cross-sectional area is \displaystyle \pi r^2 = \pi (2 + y - y^2)^2. The height of each cylinder is the same as the width of each rectangle, \displaystyle dy.

    Therefore the volume of each cylinder is \displaystyle \pi (2 + y - y^2)^2\,dy.

    So the entire volume can be approximated by \displaystyle \sum{\pi(2+y-y^2)^2\,dy}, and when you increase the number of cylinders, this sum converges on an integral and the approximation becomes exact.

    So \displaystyle V = \int_0^2{\pi (2 + y - y^2)^2\,dy}.

    Thank you again for helping me with this type of problem. I'm glad you didn't get confused at the beginning where it said x = 1 + y^2 y = x - 3. The comma I typed in somehow became a superscript so it didn't separate the two equations. The corrected way should be x = 1 + y^2, y = x - 3 but it doesn't matter because you got the equation right anyways. Kudos to you.
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  4. #4
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    Wait a minute, are you sure that is the correct setup? I thought you had to set it up like this(disk method),
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  5. #5
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    Quote Originally Posted by florx View Post
    Wait a minute, are you sure that is the correct setup? I thought you had to set it up like this(disk method),
    Read what I wrote. Are there any mistakes in my logic? (Seriously, it helps having someone else proof-read).

    Don't just blindly apply formulae, logic is always more powerful.
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