How do you find the arc length of the curve $y = \frac{5 \sqrt{7}}{3} x^{\frac{3}{2}} - 9$ $y=(5sqrt7)/3x^(3/2)-9$ over the interval [0,5]?

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How do you find the arc length of the curve $y = \frac{5 \sqrt{7}}{3} x^{\frac{3}{2}} - 9$ $y=(5sqrt7)/3x^(3/2)-9$ over the interval [0,5]?

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Answer 1 · 2018-06-16T19:18:42

$s = \int_{0}^{5} \sqrt{1 + \frac{175}{4} x} . d x = \frac{1}{525} (879^{\frac{3}{2}} - 8)$ $s=int_0^5sqrt(1+175/4x)color(white).dx=1/525(879^(3/2)-8)$

Explanation:

The arc length of a function $y$ $y$ on the interval $[a, b]$ $[a,b]$ is given by:

$s = \int_{a}^{b} \sqrt{1 + {(\frac{d y}{d x})}^{2}} . d x$ $s=int_a^bsqrt(1+(dy/dx)^2)color(white).dx$

Here:

$y = \frac{5 \sqrt{7}}{3} x^{\frac{3}{2}} - 9$ $y=(5sqrt7)/3x^(3/2)-9$

$\frac{d y}{d x} = \frac{5 \sqrt{7}}{3} (\frac{3}{2} x^{\frac{1}{2}}) = \frac{5 \sqrt{7}}{2} x^{\frac{1}{2}}$ $dy/dx=(5sqrt7)/3(3/2x^(1/2))=(5sqrt7)/2x^(1/2)$

Then the arc length desired is:

$s = \int_{0}^{5} \sqrt{1 + {(\frac{5 \sqrt{7}}{2} x^{\frac{1}{2}})}^{2}} . d x$ $s=int_0^5sqrt(1+((5sqrt7)/2x^(1/2))^2)color(white).dx$

$s = \int_{0}^{5} \sqrt{1 + \frac{175}{4} x} . d x$ $s=int_0^5sqrt(1+175/4x)color(white).dx$

Let $u = 1 + \frac{175}{4} x$ $u=1+175/4x$ , which implies that $d u = \frac{175}{4} d x$ $du=175/4dx$ . Moreover, note the change of bounds of the integral under this substitution: $x = 0 \Rightarrow u = 1$ $x=0=>u=1$ and $x = 5 \Rightarrow u = \frac{879}{4}$ $x=5=>u=879/4$ . The integral is then:

$s = \frac{4}{175} \int_{1}^{879 / 4} u^{\frac{1}{2}} . d u$ $s=4/175int_1^(879//4)u^(1/2)color(white).du$

Integrating using the power rule for integration:

$s = \frac{4}{175} (\frac{2}{3} u^{\frac{3}{2}}) {∣ ∣ ∣}_{1}^{879 / 4}$ $s=4/175(2/3u^(3/2))|_1^(879//4)$

$s = \frac{8}{525} ({(\frac{879}{4})}^{\frac{3}{2}} - 1^{\frac{3}{2}})$ $s=8/525((879/4)^(3/2)-1^(3/2))$

$s = \frac{8}{525} (\frac{879^{\frac{3}{2}}}{8} - 1)$ $s=8/525(879^(3/2)/8-1)$

$s = \frac{1}{525} (879^{\frac{3}{2}} - 8)$ $s=1/525(879^(3/2)-8)$

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How do you find the arc length of the curve $y = \frac{5 \sqrt{7}}{3} x^{\frac{3}{2}} - 9$ $y=(5sqrt7)/3x^(3/2)-9$ over the interval [0,5]?

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How do you find the arc length of the curve $y = \frac{5 \sqrt{7}}{3} x^{\frac{3}{2}} - 9$ $y=(5sqrt7)/3x^(3/2)-9$ over the interval [0,5]?