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Integral with functions of two variablesHelp!

  1. Oct 30, 2012 #1
    I really need help with this problem.

    Find the volume of the solid enclosed by the surface z=1+exsiny and the planes x=±1, y=0, y=pi and z= 0


    My logic tells me that
    we have some rectangles R[-1,1]x[0,pi] but I'm confused by the z=0
     
  2. jcsd
  3. Oct 30, 2012 #2

    LCKurtz

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    The planes ##x=\pm 1, y=0, y=\pi## form the vertical side walls of the solid. The plane ##z=0## forms the "floor" or base of the solid. The equation ##z=e^x\sin y+1## is the curvy roof of the solid. Does that help?
     
    Last edited: Oct 30, 2012
  4. Oct 30, 2012 #3
    This gives me a better picture of what is going on :)
    So I will double integrate the function z, first with respect to x in the inverval [-1,1] and then with respect to y in the interval [0,2]

    I guess there is no reason to include z=0, or do i somehow have to include that plane in the equation as well?
     
  5. Oct 30, 2012 #4

    LCKurtz

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    It is automatically included in that formula for volume:$$
    \iint_A (z_{upper} - z_{lower})\, dydx$$In this case ##z_{lower}= 0##.
     
  6. Oct 30, 2012 #5
    Ahh i see. so upper is z=e^x*sin(y) +1 and if we integrate it with respect to x we get ex*sin(y)+x because y is a constant so y is a constant right? :) and then we insert upper and lower limit (-1,1) and do the same for y with limits [0,2]
     
  7. Oct 30, 2012 #6

    LCKurtz

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    Yes.
     
  8. Oct 30, 2012 #7
    When i integrate with respect to x first with limit [-1,1] i get e*sin(y)-e-1*sin(y)

    Jesus i gotta lern to write in latex!! '
    if i type ∫ e*sin(y)-e-1*sin(y), y,0,2) into my calculator i get 3,32 does that sound right. And do you know of any way to type the whole ting into a ti89 with limits for both?

    Btw. Thank you for helping out!
     
  9. Oct 30, 2012 #8

    LCKurtz

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    That looks correct, but why would you want to use a calculator in the first place? Can't you just leave the exact answer in terms of ##e## and ##\cos 2##?
     
  10. Oct 30, 2012 #9
    Yes I can and finally did it all by hand!
    It always feels rewarding when you succeed in doing the things you find most difficult :)
     
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