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Thread: exact length of curve

  1. #1
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    exact length of curve

    First, here's the problem:

    A curve is given by $\displaystyle y=(9-x^\frac{2}{3})^\frac{3}{2} ; 1\leq y\leq 8 $. Find the exact length of the curve analytically by antidifferentiation.

    First I found the derivative of $\displaystyle y=(9-x^\frac{2}{3})^\frac{3}{2}$, then I tried to integrate it from 1 to 8. The answer I got is 10.6712.

    Can anyone confirm this answer?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by linlinrocks View Post
    First, here's the problem:

    A curve is given by $\displaystyle y=(9-x^\frac{2}{3})^\frac{3}{2} ; 1\leq y\leq 8 $. Find the exact length of the curve analytically by antidifferentiation.

    First I found the derivative of $\displaystyle y=(9-x^\frac{2}{3})^\frac{3}{2}$, then I tried to integrate it from 1 to 8. The answer I got is 10.6712.

    Can anyone confirm this answer?
    You need to find the derivative, square it, and evaluate $\displaystyle L=\int_a^b\sqrt{1+\left(\frac{\,dy}{\,dx}\right)^2 }\,dx$

    Since $\displaystyle y=\left(9-x^\frac{2}{3}\right)^{\frac{3}{2}}$, it follows that $\displaystyle y^{\prime}=\frac{3}{2}\left(9-x^{\frac{2}{3}}\right)^{\frac{1}{2}}\cdot-\tfrac{2}{3}x^{-\frac{1}{3}}$

    Squaring the derivative, we have $\displaystyle \left(y^{\prime}\right)^2=x^{-\tfrac{2}{3}}\left(9-x^{\frac{2}{3}}\right)=9x^{-\frac{2}{3}}-1$

    Now, we see that $\displaystyle L=\int_1^8\sqrt{1+\left(9x^{-\frac{2}{3}}-1\right)}\,dx=\int_1^8\sqrt{9x^{-\frac{2}{3}}}\,dx=\int_1^8 3x^{-\frac{1}{3}}\,dx$

    We see that this becomes $\displaystyle \tfrac{9}{2}\left.\left[x^{\frac{2}{3}}\right]\right|_1^8=\tfrac{9}{2}\left(8^{\frac{2}{3}}-1^{\frac{2}{3}}\right)=\tfrac{9}{2}\left(4-1\right)=\tfrac{27}{2}$

    So the arclength should be $\displaystyle \boxed{L=\tfrac{27}{2}}$.

    Does this make sense?
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  3. #3
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by linlinrocks View Post
    First, here's the problem:

    A curve is given by $\displaystyle y=(9-x^\frac{2}{3})^\frac{3}{2} ; 1\leq y\leq 8 $. Find the exact length of the curve analytically by antidifferentiation.

    First I found the derivative of $\displaystyle y=(9-x^\frac{2}{3})^\frac{3}{2}$, then I tried to integrate it from 1 to 8. The answer I got is 10.6712.

    Can anyone confirm this answer?
    You need to know the correct formula.
    If you differentiate and then integrate a function you just end up back where you started,
    so clearly that cannot be the formula for arc length.
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