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Find the length of the curve defined by $y=6 x^{3/2} - 7$ from $x=1$ to $x = 9$.

I need help with this section. I would really appreciate it. Thank you!

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    @DavidMitra, Yes, thank you (:2012-12-17

3 Answers 3

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Use the formula

$ s = \int_{a}^{b}\sqrt{1+y'(x)^2} dx $

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    @Ceelos: You are welcome.2012-12-17
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Hint: Compute $y'$ and then compute the length $\mathcal{l}$:

$\mathcal{l} =\int_1^9 \sqrt{1 + (y'(x))^2}dx$

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We have $ y=6x^{(3/2)} -7$ Let $\mathcal{L}$ denote the length of the curve. We can calculate $\mathcal{L}$ by \begin{align} \mathcal{L} &= \int_1^9 \sqrt{1+(y')^2} \, dx \\ &= \int_1^9 \sqrt{1+((6x^{(3/2)} -7)')^2} \, dx \\ &= \int_1^9 \sqrt{1+(9x^{(1/2)})^2} \, dx \\ &= \int_1^9 \sqrt{1+81x} \, dx \\ &= \Bigl[\frac{2}{243} (1+81x)^{3/2} \Bigr]_1^9 \approx 156.22 \end{align} See Wolfram Alpha for the integration.