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One integral

  1. Dec 9, 2012 #1
    1. The problem statement, all variables and given/known data
    Solve double integral
    [tex]\int^1_0\int^1_x\sin(y^2)dydx[/tex]


    2. Relevant equations



    3. The attempt at a solution
    I got with Wolfram Mathematica 7.0 result 0.23 numerically. Can it be solved analyticaly?
     
  2. jcsd
  3. Dec 9, 2012 #2

    SammyS

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    Yes, it can be solved analytically. Change the order of integration.
     
  4. Dec 10, 2012 #3
    I'm not sure how?
     
  5. Dec 10, 2012 #4
    At present, you're integrating over a set of points (x,y) with

    (1) 0 ≤ x ≤ 1
    (2) x ≤ y ≤ 1

    If you're going to reverse the order of integration, you need two new restraints:

    (1') (some number) ≤ y ≤ (some other number)
    (2') (some number or an expression with y) ≤ x ≤ (another expression that may contain y)

    Try sketching this set on a piece of paper and translate (1),(2) to a geometric shape and that back again to (1'),(2'). Then you can rewrite your integral:
    $$
    \int_{(1)}\int_{(2)} \sin(y^2)\, \mathrm dy \, \mathrm dx = \int_{(1')}\int_{(2')}\sin(y^2) \, \mathrm dx \, \mathrm dy
    $$
     
  6. Dec 10, 2012 #5

    Dick

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    Draw the region you are integrating over. It's a triangle in the xy plane, right? Then just set the integration up so you do dx first then dy.
     
  7. Dec 10, 2012 #6

    Mark44

    Staff: Mentor

    To add to what Dick said, whenever a situation arises where you're considering changing the order of integration, it's alway a good idea to sketch the region over which integration is taking place.
     
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