1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Triple integral in cylindrical coordinates

  1. Mar 30, 2015 #1
    1. The problem statement, all variables and given/known data
    Evaluate ## \int \int \int_E {x}dV ## where E is enclosed by the planes ##z=0## and ##z=x+y+5## and by the cylinders ##x^2+y^2=4## and ##x^2+y^2=9##.

    2. Relevant equations

    ## \int \int \int_E {f(cos(\theta),sin(\theta),z)}dzdrd \theta ##
    How do I type limits in for integration?

    3. The attempt at a solution
    Right now I'm just trying to find the limits of integration.

    For dz, ##z=x+y+5## is equivalent to ##z=rcos \theta +r sin \theta +5## so that is the upper limit for ##z## while ##z=0## is the lower limit.

    For dθ I am going to assume that it is ##0 < \theta < 2 \pi ## By the way, how do I make the "greater than or equal to" sign in Latex? I choose zero and two pi because question didn't say the object is restricted to any octant. Keep in mind that I do not know what the graph looks like visually so I am taking a risk...

    For dr, I have ##r^2 = 4## and ##r^2 = 9## what do I do? Additionally, can the lower limit for r ever be negative?
     
  2. jcsd
  3. Mar 30, 2015 #2

    Mark44

    Staff: Mentor

    In polar and cylindrical coordinates you're going to need an extra factor of r, as in ##r~dr~d\theta##, which is equivalent to dx dy in rectangular coordinates.
    # # \int_{a}^{b} ... # #
    Here a is the lower limit and b is the upper limit. If either limit is a single character, you don't need the braces. However, if a limit consists of two or more characters, like -3, then you need the braces.
    \ge for ≥ and \le for ≤
    Then you should sketch a graph of your region. It's not that complicated. The region is a hollow tube whose walls are formed by the two cylinders, with the cap being the plane z = x + y + 5. This plane makes a diagonal slice through the tube.
    You can take r = 2 and r = 3. It's possible for r to be negative, due to the ambiguity of coordinates in polar and cylindrical coordinates. By that I mean that a single point can have multiple representations, which is not the case in rectangular coordinates. For example (-1, 7π/6) is the same point as (1, π/6).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Triple integral in cylindrical coordinates
Loading...