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Double integration Intervals

  1. Sep 2, 2003 #1
    I'm currently taking multivariable calculus and we're studying double and triple integration. The actual integration itself is easy, but what I am not understanding at all is how to find the intervals for integration. I can only manage to find them on only the most simple problems. I don't need help solving the integral, just help finding the limits of integration. At least I hope I'm doing the integration correctly...

    Example 1
    Find the volume of the solid bounded by the cylinder x^2+z^2=9 and the planes x=0, y=0, z=0, x+2y=2 in the first octant.


    For this problem, all variables must be positive, so, 0<=x<=3 and 0<=y<=1-x/2. This doesn't appear to yield the correct answer however...On this one I integrated [squ](9-x^2). Is that the correct function to integrate? The right answer for this one is (1/6)(11[squ]5-27)+(9/2)arcsin(2/3). I get something relatively close but ultimately incorrect.

    Example 2
    Use a double integral to find the area of the region enclosed by the lemniscate r^2=4cos(2theta).

    On this I assumed the interval of r would be from 0 to when cos(2theta) was 1, so 0<=r<=4 and 0<=theta<=2pi. This doesn't yield the correct answer either, however. The right answer on this one is 4, I get 0.

    Any help at all is greatly appreciated. Note that I don't need you to solve the problems for me, just help me understand how to compute the intervals. Thanks.
     
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  3. Sep 2, 2003 #2

    Hurkyl

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    For example 1, consider what the interval of allowed values for y is when x = 3...


    For the lemniscate, I suggest sketching its graph.
     
  4. Sep 2, 2003 #3

    HallsofIvy

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    Your problem is that you are trying to use constants for both integrals: that's the same as integrating over a rectangle, not the figures you want.

    1. Find the volume of the solid bounded by the cylinder x^2+z^2=9 and the planes x=0, y=0, z=0, x+2y=2 in the first octant.

    Okay, geometrically, this is a quarter of the cylinder (with axis along the y-axis) bounded below by y=0 and above by y= 1- x/2. (Which meets y= 0 at x= 2 so does not actually include the whole cylinder: that has radius 3.)

    Draw picture (I am extremely "drawing impaired" but I can draw "side views" involving only two axes at a time). Exactly what the limits of integration are depends upon the order of integration chosen. It's an interesting exercise to shift from one to another.

    I think I will choose to write the volume as int(int (int dy)dz) dx
    The "outer integral" is "dx" and so must have numerical limits: looking at my graph I see that x can range from x= -3 (the boundary of the cylinder) to x= 2 (where x+ 2y= 2 intersects y= 0).
    For each value of x, z satisfies x^2+ z^2= 9 -> z^2= 9- x^2 or
    z= +/- sqrt(9- x^2). In the "dz" integral, for each x, z can range from -sqrt(9- x^2) to +sqrt(9- x^2) and those are the limits of integration.
    For the "inside integral- dy", the limits of integration can depend on both x and z. Here, because y only appears in the equations y=0 and x+ 2y= 2, y can range from 0 up to 1- x/2. Those are the limits of integration on the innermost integral.
    You have int(x=-3 to 2){int(z=-sqrt(9-x^2) to sqrt(9-x^2){int(y= 0 to 1-x/2) dy} dz} dx.

    2.Use a double integral to find the area of the region enclosed by the lemniscate r^2=4cos(2theta).
    (Interesting figure. Assuming you want the area of both lobes, you need to let theta range from 0 to 2 pi.)
    Here, of course, we will need to use polar coordinates. The "differential of area" in polar coordinates is r dr dtheta.

    In order to get the entire figure we need to let theta range from 0 to 2 pi. (I mention that because, sometimes with that "2 theta" term, we find that just 0 to pi covers the figure. Going from 0 to 2pi covers twice and gives twice the area.)
    FOR EACH THETA, r ranges from 0 out to the graph: r= sqrt(4cos(2theta)): notice we DON'T NEED r= - sqrt(4 cos(2theta)) in polar coordinates, r is always +, using negative values just reflects back the other way and we are already getting that.

    The integral is int(theta= 0 to 2pi){int(r= 0 to sqrt(4cos(2theta)) r dr} dtheta.
     
  5. Sep 2, 2003 #4
    Halls, on that first integral, what function do you integrate? For that particular problem I don't think we're supposed to use triple integration.

    So since y is negative when x=3, the max for x is 2? Why is x allowed to be negative and not be bound by x=0?

    Edit: I just changed the interval from -3<=x<=2 to 0<=x<=2 and got the right answer! Thanks a lot guys.

    The second function is still puzzling me however..
     
  6. Sep 2, 2003 #5

    Hurkyl

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    Did you try sketching the graph for problem two? The equation has a distinctive property... for example, consider what happens when &theta;=&pi;/4.
     
  7. Sep 2, 2003 #6

    HallsofIvy

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    The way I did it, since you asked for volume, there is no "function". Volume is the triple integral of dydzdx. Of course, you could go ahead and do the first integral: int(y= 0 to 1-x/2) dy
    is 1-x/2 so the integral is int(x= 0 to 2) int(z= 0 to sqrt(9-x^2) dz dx.

    That's right.
    'Cuz I messed up! I was focusing on the "cylinder" and forgot about the "x= 0, y= 0, z=0" part! The integrals should be
    int(x= 0 to 2) int(z= 0 to sqrt(9-x^2)) int(y=0 to 1-x/2) dydzdx.
     
  8. Sep 2, 2003 #7
    It's the infinity sign, and at &pi;/4 the value of the function is zero..

    Using -&pi;/4 to &pi;/4 as the interval, which is one lobe, I get 2, which is half the right answer. I can't seem to get it to be 4 however...if I make it 0 to 2&pi; I get a really small number approaching zero and using -&pi;/4 to 5&pi;/4 I get 2 also.
     
    Last edited: Sep 2, 2003
  9. Sep 3, 2003 #8

    Hurkyl

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    Both lobes should have the same area, right? So the total area is 4!


    Like the other problem, you can't simply integrate from 0 to 2&pi; because some of those values are illegal. If you didn't want to use symmetry to determine the total area from knowing the area of a single lobe, I know of two ways to do it... (symmetry is still the best way, though):

    (a) The total area is the sum of two integrals; one from -&pi;/4..&pi;/4 and the other from 3&pi;/4..5&pi;/4 (you can't integrate between &pi;/4 and 3&pi;/4 because the graph doesn't exist on that interval!)

    (b) Instead of integrating the length from the origin to r for each &theta;, you could instead be integrating the length from -r to r, thus getting both lobes in your integral from -&pi;/4..&pi;/4.
     
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