## happykiddo one year ago Explain how to get the arclength of y=3/8(x^(4/3)-2x^(2/3)) from x = 1 to x = 8. I know the answer is 63/8, but don't understand how to get there.

1. IrishBoy123

$y=\frac{3}{8} (x^{4/3}-2x^{2/3})$ $1 \lt x \lt 8$ ????????

2. IrishBoy123

if this is correct, i'd go for a sub, $$p = x^{\frac{2}{3}}$$ so $$y = \frac{3}{8} (p^2 -2 p) \rightarrow dy = \frac{3}{8} (2p .dp -2 dp) = \frac{3}{4}.dp.(p-1)$$ $$x = p^{\frac{3}{2}}, dx = \frac{3}{2}p^\frac{1}{2} .dp$$ and so $ds = \sqrt{dx^2 + dy^2} \ = \sqrt{ \frac{9}{4}p + \frac{9}{16}(p-1)^2 \ } \ \ dp$

3. happykiddo

Could you explain the u-substitution that you use here? Why you chose the "p" you did?Why use u-sub twice?....ETC......Or is there a rule, this falls under? Like always your help is much appreciated.

4. happykiddo

Did you get 63/8, because I got 37.

5. anonymous

$y=\frac{ 3 }{ 8 }(x ^{4/3}-2x ^{2/3})$ $y=\frac{ 3 }{ 8 }x ^{4/3}-\frac{ 3 }{ 4 }x ^{2/3}$

6. anonymous

$\frac{ dy }{ dx }=\frac{ 1 }{2 }x ^{1/3}-\frac{ 1 }{ 2 }x ^{-1/3}$

7. anonymous

$\frac{ dy }{ dx }=\frac{ 1 }{ 2 }x ^{1/3}-\frac{ 1 }{2 }x ^{-1/3}$

8. anonymous

Can you screenshot me the problem?

9. IrishBoy123

@happykiddo the sub comes from messing around with it for a bit. those rational exponents just look really annoying, especially as you have the $$\sqrt{}$$ to deal with. sorry i can't be more precise. i guess if you do enough of these, you will just get a feel for it and the answer is 63/8 as advertised. the limits become $$1 \le p \le 4$$, right?

10. IrishBoy123

$\sqrt{ \frac{9}{4}p + \frac{9}{16}(p-1)^2 \ }$ $=\sqrt{ \frac{9}{4}p + \frac{9}{16}(p^2- 2p +1) \ }$ $=\sqrt{ \frac{9}{16}(p^2- 2p +4p +1) \ }$ $=\sqrt{ \frac{9}{16}(p +1)^2 \ }$ $\implies \int\limits_{p=1}^{4}\frac{3}{4}(p+1) \ dp$

11. IrishBoy123

but you don't actually need the sub because $$dy = \frac{1}{2}(x^{\frac{1}{3}} - x^{\frac{-1}{3}})dx$$ $$\sqrt{dx^2 + dy^2} = \sqrt{1 + \frac{1}{4}(x^\frac{2}{3}+x^\frac{-2}{3}-2)} \times \ dx$$ which eventually becomes $$= \frac{1}{2} \sqrt{(x^\frac{1}{3} + x^\frac{-1}{3})^2} \times dx$$

12. happykiddo

Thanks for the help guys! Chrome on my computer crashed, and man was it a journey to get it working again. Anyway I understand, and appreciate the help.