1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Simple area between curves question

  1. Jan 1, 2007 #1
    1. The problem statement, all variables and given/known data
    Find the area of the region enclosed by the following curves:

    2. Relevant equations

    3. The attempt at a solution
    I'm confused by the graph because the region enclosed has positive and negative parts, and I can't determine whether f(y)>g(y), g(y)>f(y), or what. I'm not sure what to integrate here.

    Thanks, I appreciate the help.
  2. jcsd
  3. Jan 1, 2007 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You could use the modulus function;

    [tex]A = \int^{a}_{b} \left\|f(y)\right\| + \left\|g(y)\right\|\;dy[/tex]

    Also, note that f(y) = -g(y)
  4. Jan 1, 2007 #3

    Gib Z

    User Avatar
    Homework Helper

    The Solution to the integral [tex]\int f(y) g(y) dy[/tex] is zero, as f(y)=-g(y).

    You could find the definite integral of each, absolute valued. Or a simpler method would be to use symmetry arguments to realize the solution is [tex]2\int^{1}_{-1} 1-y^2 dy[/tex], which you should be fine with.
    Last edited: Jan 1, 2007
  5. Jan 1, 2007 #4
    Are you sure that integral is zero? You might want to recheck your math unless you meant that product to be a sum.
  6. Jan 1, 2007 #5

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    The curves meet at +1 and -1. You need the area between the curves as y goes from +1 to -1. I'm not sure why you can't tell that f(y) =>g(y) in that region: f(0)=1 and g(0)=-1, so it is clear which is 'on top'. The area is just the integral of f(y)-g(y); I don't mind telling you this (we aren't supposed to just hand out answers) since it is just what you were told in class/in the book. It doesn't matter whether f or g are positive or negative individually: you only care about one relative to the other, the actual signs of f and g don't matter.

    For any functions f, and g, the area bound between them between a and b is always

    [tex] \int_a^b \max(f(t),g(t)) - \min(f(t),g(t))dt[/tex]

    a=-1, b=1 here, and f(t)=>g(t) for all t in [-1,1].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook