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Math Help - In need of verification and help, integration of volumes

  1. #1
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    In need of verification and help, integration of volumes

    Hi. I am not sure if these questions have been posted before. Reworked them today, the answers are not in the book:

    Find volume generated by revolving the regions bounded by given curves about y axis
    Question 26-3-20
    y^2 = x, y = 4, x = 0, Disks

    Attempt:
    dV = pix^2
    x^2=y^4
    Integration: pi(1/5)(4^5) = 634.4 ?

    Question 26-3-22
    y = 3x^2 - x^3, y = 0, Disks

    Attempt:
    dV = pix^2
    (3x^2 - x^3)(3x^2 - x^3) =
    6x^4 - 6x^5 + x^6
    Intercepts: x = 3 when y = 0
    Integration:
    pi 6(3^4) - 6(3^5) + 3^6 =
    pi (486-1458+729) = -763.4 ?
    negative answer is not good

    Question 26-3-24
    x^2 + 4y^2 = 4, Disks

    Attempt:
    Isolating x: x^2 = 4 - 4y^2
    x^2 = 4y - 1/3(4(y^3)) ?
    Please let me know if this is correct so far.

    Thanks for your time.
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  2. #2
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    Re: In need of verification and help, integration of volumes

    Hey togo.

    Q1 looks right.

    For Q2, what are the limits? (Are you bounding the area by some other function?)

    For Q3, try setting up your limits (between -1 and +1) and see you how go.
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  3. #3
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    Re: In need of verification and help, integration of volumes

    All the info I have has been posted, for Q2 I don't understand why no limit was given either.
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  4. #4
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    Re: In need of verification and help, integration of volumes

    For Q2 you didn't calculate the anti-derivative properly: it should be:

    F(x) = 6x^5/5 - x^6 + x^7/7

    This gives an integral value of 6/5*3^6 - 3^6 + 3^7/7 =

    > 6/5*3^6 - 3^6 + 3^7/7
    [1] 458.2286

    Can you try Q3?
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  5. #5
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    Re: In need of verification and help, integration of volumes

    Tried Q3, the question also says Quadrant 1.
    I was looking for x intercepts in order to find y boundary values, but end up dividing by zero.

    y = 1 - (x^2)/4
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  6. #6
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    Re: In need of verification and help, integration of volumes

    Show us all of your working.
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  7. #7
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    Re: In need of verification and help, integration of volumes

    Sorry the original equation I gave (y = 1 - (x^2)/4) was wrong (needs a square root)

    4y^2 = 4 - x^2
    y^2 = 1 - (x^2)/4
    y = (1 - x^2/4)^(1/2)

    is just a reworking of the problem equation algebraically in order to find the x intercepts and therefore find the y boundary to plug them into the integral.

    the integral which I cannot even try using yet is x^2 = 4y - 1/3(4y^3) although not proven correct.

    no boundaries were given in the question except "quadrant 1"
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  8. #8
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    Re: In need of verification and help, integration of volumes

    If its quadrant 1, then your integral will be bound to x = -2 and x = 0.

    Using this hint, can you evaluate the integral?
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  9. #9
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    Re: In need of verification and help, integration of volumes

    Did you graph the function in problem 2? If you do you will see what region it is that is rotated around the y-axis. You will need to do two "disk method" integrations.
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  10. #10
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    Re: In need of verification and help, integration of volumes

    damn. (Q3, working on Q2 soon)

    This is what I have:
    x^2 = 4y - 1/3(4y^3)

    principle formula dV = pix^2

    Tempted to plug -2 into y in the above integral
    but we plug in x as y?
    4(-2) - 1/3(4(-2^3))?
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  11. #11
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    Re: In need of verification and help, integration of volumes

    I still get a weird answer for integral on Q2

    6x^4 - 6x^5 + x^6
    1/5(6x^5) - 1/6(6x^6) + 1/7(x^7)
    291.6 - 729 + 312.43 = weird negative answer!
    Last edited by togo; March 4th 2013 at 10:50 AM.
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  12. #12
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    Re: In need of verification and help, integration of volumes

    If you were to graph y= 3x^2- x^3, you would see that there is a loop from (0, 0) up to (2, 4) and down to (3, 0). Rotating that around the y-axis gives a "doughnut" shaped figure- there is a hole in the middle. If you are using the "shell method" (though you said in your first post "Disks"), the radius of a disk would be x and the height y so you would have 2\pi \int_0^3 xydx= 2\pi \int_0^3 3x^3- x^4 dx.

    If you really want to use the "disk" method the radius should be x in terms of y and I don't see any good method of solving y= 3x^2- x^3 for x.
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  13. #13
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    Re: In need of verification and help, integration of volumes

    Yes shell method, not sure how disks got in there. I plotted it out and can see it now. Although I don't see any hole in the middle

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