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Homework Help: Finding the length of an arc of a parabola

  1. Nov 18, 2013 #1
    1. The problem statement, all variables and given/known data

    y^2 = x from (0,0) to (1,1)

    2. Relevant equations

    L = ∫√(1+[g'(y)]^2) dy

    3. The attempt at a solution

    So this is a problem in my textbook that has been bothering me because I can't seem to come up with the same answer.

    1. [bounds 0 to 1] 1/2∫ sec^3θ was obtained using trig substitution with y = 1/2tanθ and dy=1/2sec^2θ which, according to the steps in the textbook, is correct.

    2. I use integration by parts which gives me 1/2∫ sec^3θ = 1/2secθtanθ - 1/2∫sec^3θ + 1/2∫secθ
    adding - 1/2∫sec^3θ to the other side of the equation it becomes
    1/2secθtanθ + 1/2∫secθ=
    1/2secθtanθ + 1/2 ln(secθ + tanθ)

    then I use y = 1/2tanθ to change my bounds from 0 to θ
    and then evaluating for tanθ = 2 I get the answer

    L = √5 + ln(√5 +2) / 2

    where the book comes up with

    L = √5/2 + ln(√5 +2) / 4

    and one of it's steps after integration by parts shows

    1/4secθtanθ + 1/4 ln(secθ + tanθ)

    and I seem to be having trouble how they came up with 1/4 instead of 1/2. Most likely a stupid mistake I made and am overlooking? Thanks in advance for the help! xo <3
  2. jcsd
  3. Nov 18, 2013 #2


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    Science Advisor
    Homework Helper

    Yes. It's simple. You got 1/2∫ sec^3θ = 1/2secθtanθ - 1/2∫sec^3θ + 1/2∫secθ. That tells you ∫ sec^3θ = 1/2secθtanθ + 1/2∫secθ. Which is fine. But you wanted to integrate (1/2)∫ sec^3θ. It's a little confusing with all of the (1/2)'s running around. Easy to miss one.
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