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Math Help - help finding exact arc length of a curve

  1. #1
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    help finding exact arc length of a curve

    I need help with the following problem:

    Find the exact arc length of the curve y= 2/3 * (x+1)^3/2 where the interval for x is [2,7]. Do the algebra and calculus carefully showing your work. (Final answer should be 38/3).

    Here is what I tried to do, but was not able to get a final answer of 38/3:

    First I found the derivative: y'=2/3(3/2)(x+1)^1/2 *1 = 1(x+1)^1/2

    Then I plugged it into the formula for arc length:

    s= ∫[from 2 to 7]Sqrt of 1 + ((x+1)^1/2)^2 dx
    s= ∫[from 2 to 7]Sqrt of 1 + x^5/2 + 1^5/2 dx
    s= ∫[from 2 to 7]Sqrt of (1+ x^5/2 + 1)^1/2
    s= ∫[from 2 to 7]Sqrt of (1+ x^3 + 1)
    s= (2 + x^3) | from 2 to 7
    s=(2 + 7^3)-(2 + 2^3)
    s=345-10
    s=335

    I'm guessing I must have messed up somewhere since my answer isn't anywhere close to 38/3. I am not sure where I messed up or what I did wrong, but if anyone can explain to me how to do this problem or explain what I am doing wrong I would greatly appreciate it. Thanks in advance to anyone who can help.
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  2. #2
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    Just some computation stuff
    L= \displaystyle\int^7_2 \sqrt{1+ (x+1)^{1/2\cdot2}}\,dx =  \displaystyle\int^7_2 \sqrt{x+2}\,dx
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Matt164 View Post
    I need help with the following problem:

    Find the exact arc length of the curve y= 2/3 * (x+1)^3/2 where the interval for x is [2,7]. Do the algebra and calculus carefully showing your work. (Final answer should be 38/3).

    Here is what I tried to do, but was not able to get a final answer of 38/3:

    First I found the derivative: y'=2/3(3/2)(x+1)^1/2 *1 = 1(x+1)^1/2

    Then I plugged it into the formula for arc length:

    s= ∫[from 2 to 7]Sqrt of 1 + ((x+1)^1/2)^2 dx
    s= ∫[from 2 to 7]Sqrt of 1 + x^5/2 + 1^5/2 dx
    s= ∫[from 2 to 7]Sqrt of (1+ x^5/2 + 1)^1/2
    s= ∫[from 2 to 7]Sqrt of (1+ x^3 + 1)
    The next line does not follow from the above line. (and note what scopur says as well)

    s= (2 + x^3) | from 2 to 7
    s=(2 + 7^3)-(2 + 2^3)
    s=345-10
    s=335

    I'm guessing I must have messed up somewhere since my answer isn't anywhere close to 38/3. I am not sure where I messed up or what I did wrong, but if anyone can explain to me how to do this problem or explain what I am doing wrong I would greatly appreciate it. Thanks in advance to anyone who can help.

    CB
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  4. #4
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    Thanks. I see what I did wrong now. However, I am still not able to get the answer of 38/3. When I compute the integral: ∫[from 2 to 7]sqrt of (x+2)

    I get: sqrt of(7+2) - sqrt of(2+2)= 3 - 2 = 1

    Is 1 the correct answer or did I do something else wrong? I'm thinking maybe 38/3 is wrong and the book made a mistake or something. If anyone can help me I would appreciate it very much. Thanks.
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  5. #5
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    Does anyone know why I'm getting an answer of 1 instead of 38/3? If anyone knows why I am getting this answer, please help. Thanks.
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  6. #6
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    L= \displaystyle\int^7_2 \sqrt{1+ (x+1)^{1/2\cdot2}}\,dx =  \displaystyle\int^7_2 \sqrt{x+2}\,dx

     = \left[\frac{2}{3}(x+2)^{\frac{3}{2}}\right]^7_2 = \frac{2}{3}(9)^{\frac{3}{2}}-\frac{2}{3}(4)^{\frac{3}{2}}= 18-\frac{16}{3} = \frac{38}{3}
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