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Homework Help: Trig Substitution and Volume of Solids

  1. Nov 9, 2011 #1
    2 Questions here! (I'm not exactly sure if it's allowed, but I want to avoid posting too many threads)

    1. The problem statement, all variables and given/known data
    1)Find the following definite integrals by using a trigonometric substitution:

    2. Relevant equations

    3. The attempt at a solution
    From a previous question, I found that the indefinite integral of ∫dx/(√(2x-x2) was arcsin(x-1) + C
    Using FTC part 2, I got:
    arcsin(1-1)-arcsin(1/2-1)=arcsin0-arcsin(-1/2)=0- -∏/6=∏/6

    The answer the teacher gave me was ∏/2
    I'm have no idea where I have made a mistake.

    1. The problem statement, all variables and given/known data
    Find the volumes of the solids generated by revolving the region bounded by the curves y=x+2 and y=x2 about (a) the line x=2 (b) the line y=4

    2. Relevant equations
    Shell:V= 2∏abxhdx
    Disc: V=∏abr22-r12dx

    3. The attempt at a solution
    x=-1 and x=2


    The correct answer was supposed to be 27∏/2

    The answer is supposed to be 108∏/5

  2. jcsd
  3. Nov 9, 2011 #2
    For d) It is not
  4. Nov 9, 2011 #3

    I like Serena

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    Homework Helper

    Welcome to PF, Luscinia! :smile:

    No mistake, your answer to (d) is correct.
    It is arcsin(x-1) + C and it evaluates to ∏/6.
    (Sorry sandy.bridge.)
  5. Nov 9, 2011 #4
    Oops, you're right. I presume that your issue was with regards to your lower and upper limits. Did you remember to alter them during substitution?
  6. Nov 9, 2011 #5
    Thanks for the quick answer!
    I'm not sure why I need to replace the upper and lower limit since the final integral for ∫dx/(√(2x-x2) uses the variable x as well.
    (Though I honestly forgot about them and I'm not too sure if I know how to do it properly)
    I had used (x-1)=sinu as substitution when I had integrated so arcsin(x-1)=arcsin(sinu)=u

    arcsin(-1/2)=u=-∏/6 (lower limit)

    arcsin(0)=u=0 (upper limit)

    using FTC:

    Maybe my teacher made a mistake?
  7. Nov 9, 2011 #6
    Okay, I just did the question and got the same answer as you. Your teacher must have given you a false answer.
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