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Math Help - How do I find the limits of integration?

  1. #1
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    How do I find the limits of integration?

    In a problem such as this:

    Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.
    y + x = 5,
    y - x = 0, y + 3x = 4

    or this:

    Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.
    x = y3 - 18y,
    y + 6x = 0
    How do I find the limits of integration?
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  2. #2
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by bigdan View Post
    In a problem such as this:

    Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.
    y + x = 5,
    y - x = 0, y + 3x = 4

    or this:

    Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.
    x = y3 - 18y,
    y + 6x = 0
    How do I find the limits of integration?
    first of all, draw your self those lines and you will see limits for integrations ...


    to draw any line you need just 2 points and connect them ... (for curves you need at least 3 points)

    so ....
     x+y = 5 \Rightarrow x = 5-y so for y = 0 you have x = 5 , and for let's say y = 5 you will have x = 0 so you have 2 points M(5,0) N(0,5) yo you draw the line through those 2 points ... do the same thing with another 2 equations and you will get that region that you need to integrate

    to see points of the intersecting, just put first equation in second and see where those 2 lines intersect , same thing with second and third and of course first and third .... and from that you will get your points of integration...

    intersecting point of the first and second
     y+x = 5 \Rightarrow y = 5-x put in the second  y-x = 0 == 5-x-x=0 \Rightarrow 2x = 5 \Rightarrow x = \frac {5}{2}
    intersecting first and third
     y+x =5 \Rightarrow y=5-x put in the third  y+3x=4 == 5-x+3x=4 \Rightarrow 2x=-1 \Rightarrow x=- \frac {1}{2}
    and second and third ... x =1



    Hint: There are two regions of integration.... calculate both and add them  \displaystyle \int _? ^? dx \int _? ^? dy + \int _? ^? dx \int_? ^? dy


    P.S. second problem do the same way
    Attached Thumbnails Attached Thumbnails How do I find the limits of integration?-integration_limits.jpg  
    Last edited by yeKciM; September 21st 2010 at 11:47 PM.
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