How do you find the arc length of the curve # y = (3/2)x^(2/3)# from [1,8]?

Question

Samantha Adkison · Accepted Answer

The arc length can be quickly derived by realizing that it's just the distance formula in combination with the derivative over some interval #Deltax#. Thus: #s(x) = sum_(a)^(b) sqrt((Deltax)^2 + (Deltay)^2/(Deltax)^2(Deltax)^2)# #= sum_(a)^(b) sqrt(1 + (Deltay)^2/(Deltax)^2)Deltax# #s(x) = int_(a)^(b) sqrt(1 + ((dy)/(dx))^2)dx# Therefore: #(dy)/(dx) = 3/2*2/3x^"-1/3" = x^"-1/3"#
#((dy)/(dx))^2 = x^"-2/3"# #s(x) = int_1^8  sqrt(1 + 1/x^"2/3")dx# What normally what we would do is try to find a way to complete the square. Let's just make this look nicer for now. #= int_1^8  1/sqrt(x^"2/3")sqrt(x^"2/3" + 1)dx# #= int_1^8  1/x^"1/3"sqrt((x^"1/3")^2 + 1)dx# Now let's make the substitution:
#u = x^"1/3" => u^3 = x#
#dx = 3u^2du# #= 3int_1^8  u^2*1/usqrt(u^2 + 1)du# #= 3int_1^8  sqrt(u^2 + 1)*udu# The way I would proceed is with another substitution. Let:
#w = u^2 + 1#
#dw = 2udu#
#udu = (dw)/2# #=> 3/2 int_1^8 sqrtwdw# #= w^"3/2" = (u^2 + 1)^"3/2"# #= [(x^"2/3" + 1)^"3/2"]|_(1)^(8)# #= [(8^"2/3" + 1)^"3/2"] - [(1^"2/3" + 1)^"3/2"]# #= (4 + 1)^"3/2" - (1 + 1)^"3/2"# #= 5^"3/2" - 2^"3/2"# #= sqrt125 - sqrt8# #= color(blue)(5sqrt5 - 2sqrt2) ~~ 8.3519 "u"#

Aurora Alvarado · Answer

undefined To find the arc length of the curve $ y = \frac{3}{2}x^{\frac{2}{3}} $ from $ x = 1 $ to $ x = 8 $, we use the formula for arc length:

$$ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx $$

First, we need to find $ \frac{dy}{dx} $ which represents the derivative of $ y $ with respect to $ x $:

$$ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{3}{2}x^{\frac{2}{3}}\right) $$
$$ = \frac{3}{2} \cdot \frac{2}{3}x^{\frac{2}{3} - 1} $$
$$ = x^{-\frac{1}{3}} $$

Now, we can substitute $ \frac{dy}{dx} $ into the arc length formula:

$$ L = \int_{1}^{8} \sqrt{1 + \left(x^{-\frac{1}{3}}\right)^2} \, dx $$

$$ L = \int_{1}^{8} \sqrt{1 + x^{-\frac{2}{3}}} \, dx $$

This integral can be solved using various techniques, such as substitution or integration by parts. Once integrated, you will have the arc length of the curve over the given interval [1, 8].