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Definite integral of functions of two variables

  1. Feb 26, 2009 #1
    1. The problem statement, all variables and given/known data
    A pile of earth standing on flat ground near an abandoned mine has height 13 meters. The ground is the xy-plane; the origin is directly below the top of the pile and the z-axis is upward. The cross-section at height z is given by [tex]x^2 + y^2 = 13 - z[/tex] for [tex]0 \leq z \leq 13[/tex] with x, y, and z in meters.

    (a) What equation gives the edge of the base of the pile?
    (b) What is the area of the base of the pile?

    2. Relevant equations



    3. The attempt at a solution
    I really don't know what it means by the edge of the base of the pile, I know that after we get to part a. then the area is just the integral right?
     
    Last edited: Feb 26, 2009
  2. jcsd
  3. Feb 26, 2009 #2

    Dick

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    The 'edge of the base of the pile' is the intersection of the pile with the x-y plane. I.e. z=0. The 'area' is the area enclosed by that curve. You can find the area by integrating, but it's a pretty simple curve. You may know a formula.
     
  4. Feb 26, 2009 #3
    So the edge of the base pile is [tex]x^2 + y^2 = 13[/tex] and I just need to integrate this right.. do I integrate it over x and y? I never knew an integral rule with that formula.. I think.

    If I do an integration with respect to x and y I have:

    [tex] (x^3/3 - 13x) + (y^3/3-13y) [/tex]
     
    Last edited: Feb 26, 2009
  5. Feb 26, 2009 #4

    Dick

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    It's the area between the two functions y=+/-sqrt(13-x^2). But don't you recognize the shape of the curve?
     
  6. Feb 26, 2009 #5
    so I just integrate sqrt(13-x^2)
     
  7. Feb 26, 2009 #6

    Dick

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    You integrate 2*sqrt(13-x^2) if you want to. But I can think of easier ways to find the area of a circle.
     
  8. Feb 26, 2009 #7
    and how is that? and where did you get the 2 * in front of the sqrt(.........)
     
  9. Feb 26, 2009 #8

    Dick

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    Because you have to integrate between y=-sqrt(13-x^2) and y=+sqrt(13-x^2). You don't know how to find the area of a circle?
     
  10. Feb 26, 2009 #9
    Yes it is [tex] \pi r^2 [/tex] and the radius is [tex] \sqrt(13) [/tex] right? How come when you do the integral way you don't get a pi anywhere??
     
  11. Feb 26, 2009 #10

    Dick

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    You do get a pi. You have to do a trig subsitution to do the integral.
     
  12. Feb 26, 2009 #11
    so the area is just [tex] 13 \pi [/tex]
     
  13. Feb 28, 2009 #12
    And so if there's another question:

    What equation gives the cross-section of the pile with the plane z = 10?
    The answer is x^2+y^2 = 3 and what is the area of this?
     
  14. Feb 28, 2009 #13

    Dick

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    Do it the same way as x^2+y^2=13.
     
  15. Feb 28, 2009 #14
    and I can just use the area of a circle with radius sqrt(3) as well here to make life easier?
     
  16. Feb 28, 2009 #15

    Dick

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    Why would you think not?
     
  17. Feb 28, 2009 #16
    Just clarifying, and what is A(z), the area of a horizontal cross-section at height z?

    is it just z = - x^2 - y^2 - 13
     
  18. Feb 28, 2009 #17

    HallsofIvy

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    1. This is not a function of z so it can't be the area function you want.

    2. Since the formula for the cone is [itex]x^2+ y^2= 13- z[/itex], you want the equation
    of the circular boundary at height z, not a equation for z. In fact, [itex]x^2+ y^3=13-z[/itex] is precisely what you want!
     
  19. Feb 28, 2009 #18
    why is it z-13-z? where did you get the extra -z from? z-13-z is just 13 right?
     
  20. Feb 28, 2009 #19

    HallsofIvy

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    That was a typo. I have edited it.
     
  21. Feb 28, 2009 #20
    if I take the integral of this area then the units would be meters^3? as it would be the volume
     
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