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General Question About Trig Substitutions (integration)

  1. Oct 29, 2007 #1
    So I am trying to get a section ahead in my calsulus text and I am at Trig substitutions.

    It says, "To start we will be concerned with integrals that contain expressions of the form [tex]\sqrt {a^2-x^2}[/tex] where a is positive and real...etc"

    The idea is to eliminate the radical. For the above example they start by saying "we can make the substitution [tex]x=a\sin\theta[/tex] " ...and then they give

    absolutely no justification for using [tex]a\sin\theta[/tex]. To me that is like saying "well instead of building that house out of wood, let's use cheese instead."

    Now this is what I have reasoned out. Would someone please let me know if I am on the right track:

    Since it is the sqrt function, then the term x^2 must be less than or equal to a^2 in order to have a real solution. Since a is a positive real number, than the product a*sin(theta) must equal x for some angle theta.

    Thanks,
    Casey
     
    Last edited: Oct 29, 2007
  2. jcsd
  3. Oct 29, 2007 #2

    Dick

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    If |x|<=|a| then x can be written as a*sin(theta) for some angle theta. I don't see what is bugging you.
     
  4. Oct 29, 2007 #3
    Well, for starters, ^^^that's exactly what I wrote, isn't it:rolleyes: so that is what's bugging me right now.

    Casey

    p.s. I don't like texts who just say "this is the way it is and that's that."
     
  5. Oct 29, 2007 #4

    Dick

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    If you want to know why, then sin(theta) assumes all values between -1 and 1. So if |x|<=|a|, you can find a value of theta. It's just a change of variables.
     
  6. Oct 29, 2007 #5
    So I had it when I said:

    I assume this same concept will apply to the other trig subs as well.

    Thanks,
    Casey
     
  7. Oct 29, 2007 #6

    Dick

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    Yes, I guess I wasn't quite sure what the question was.
     
  8. Oct 29, 2007 #7
    There isn't one.

    Casey
     
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