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Integrating Areas between curves

  1. Apr 26, 2012 #1
    I can't figure how to solve this problem. Is says ∫∫(x^2)y dA where R is the region bounded by curves y=(x^2)+x and y=(x^2)-x and y=2. I can't figure how to do the limits with that. Please help!
  2. jcsd
  3. Apr 26, 2012 #2

    [itex]y=x^2+x[/itex] is an ascending straight parabola with zeroes at [itex]-1\,,\,0[/itex] , and [itex]y=x^2-x[/itex] is a similar such parabola with zeroes at [itex]0\,,\,1[/itex].

    Well, now draw both parabolas together, find their intersection points and...voila!

  4. Apr 26, 2012 #3
    I was able to get all that. I even graphed it but I can't figure out how to go about writing the integral.
  5. Apr 26, 2012 #4


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    Where do the two parabolas intersect? Where does y= 2 intersect the two parabolas?

    It looks to me that you will want to do this in two separate integrals. First, take x from the value of x where the line y= 2 intersects [itex]y= x^2+ x[/itex] to x= 0. You should be able to see that, for each x, y goes from the parabola up to [itex]y= x^2+ x[/itex]. Then do the second integral from x= 0 to the intersection of y= 2 and [itex]y= x^2- x[/itex].
  6. Apr 28, 2012 #5
    Hey man,you should do this in 2 double integrals-
    0 2
    ∫ ∫ f(x,y) dy dx
    -1 (x(x-1))

    1 2
    ∫∫ f(x,y) dy dx
    0 (x(x+1))

    and then add the 2 for your total volume.
    I hope this is right :smile:
    Last edited: Apr 28, 2012
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