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Math Help - use double integral to find volume of the solid bounded by the paraboloid & cylinder

  1. #1
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    use double integral to find volume of the solid bounded by the paraboloid & cylinder

    use double integral to find volume of the solid bounded by the paraboloid z=x^2+y^2 above, xy plane below, laterally by circular cylinder x^2 +(y-1)^2 = 1

    So, I broke it above and below y-axis, and used polar: r varies from 0 to 2sin(theta) and theta varies from 0 to pi.


    V = 2* integral from 0 to pi [[ integral from 0 to 2sin(theta) r^2 * r dr ]] dtheta

    first integral results: 8sin^4(theta), second integral results: 3pi.

    I think the answer is supposed to be 3pi/2. My limits are right??

    checked my calculations:
    integrate 2r^2 *r dr from 0 to 2sinx - Wolfram|Alpha
    integrate 8sin^4(x) from 0 to pi - Wolfram|Alpha
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    Re: use double integral to find volume of the solid bounded by the paraboloid & cylin

    The base of the solid is a circle entirely above the x-axis. So what you are doing is correct.

    \int _0^{\pi }\int _0^{2 \text{ sin}(\theta )}r^3drd\theta  = \frac{3 \pi }{2}
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    Re: use double integral to find volume of the solid bounded by the paraboloid & cylin

    So what you are saying is my limits are correct and the answer should be 3pi/2? So the results I got from wolfram are wrong, since they stipulate the answer is 3pi (I linked my entries)
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    Re: use double integral to find volume of the solid bounded by the paraboloid & cylin

    Quote Originally Posted by SNAKE View Post
    So what you are saying is my limits are correct and the answer should be 3pi/2? So the results I got from wolfram are wrong, since they stipulate the answer is 3pi (I linked my entries)
    yes the limits and the answer are correct.
    I don't see why you multiply the answer times 2 since there is no above and below (only above the x-axis) unless I am misreading the statement of the problem.
    Can you give us the link to where you found the answer 3pi ?
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    Re: use double integral to find volume of the solid bounded by the paraboloid & cylin

    3pi is what I got when I did it myself (through wolfram):
    integrate 2r^2 *r dr from 0 to 2sinx - Wolfram|Alpha
    integrate 8sin^4(x) from 0 to pi - Wolfram|Alpha.

    The reason I multiply by two is because the limit is from 0 to pi.
    The whole thing should be 0 to 2pi for the entire circle that is on the xy plane, no?
    I found the answer 3pi/2, but didn't get it myself
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    Re: use double integral to find volume of the solid bounded by the paraboloid & cylin

    Quote Originally Posted by SNAKE View Post
    3pi is what I got when I did it myself (through wolfram):
    integrate 2r^2 *r dr from 0 to 2sinx - Wolfram|Alpha
    integrate 8sin^4(x) from 0 to pi - Wolfram|Alpha.

    The reason I multiply by two is because the limit is from 0 to pi.
    The whole thing should be 0 to 2pi for the entire circle that is on the xy plane, no?
    I found the answer 3pi/2, but didn't get it myself
    The whole thing should be 0 to pi for the entire circle not 0 to 2pi
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    Re: use double integral to find volume of the solid bounded by the paraboloid & cylin

    but why is it not 0 to 2pi to go around the whole circle? 0 to pi is a semicircle, so that would be half is what ii thought.
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    Re: use double integral to find volume of the solid bounded by the paraboloid & cylin

    Quote Originally Posted by SNAKE View Post
    but why is it not 0 to 2pi to go around the whole circle? 0 to pi is a semicircle, so that would be half is what ii thought.
    There is no rule (theorem) in polar coordinates that says 0 to pi is a semicircle and 0 to 2pi describes the whole circle.
    In our situation the angle varies from 0 to pi to describe the whole circle once around as follows:
    r = 2 sin(angle)
    angle 0 gives r = 0
    angle 45 degrees gives r = sqrt(2)
    90 degrees r = 2
    150 degrees r = 1
    180 degrees r = 0
    and the circle is complete
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    Re: use double integral to find volume of the solid bounded by the paraboloid & cylin

    use double integral to find volume of the solid bounded by the paraboloid & cylinder-soru.jpg
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