Finding Arc Length of a Function

**coolio** · March 4th 2008, 07:58 AM

I'm having some trouble with these:

1. Find the arc length of the function: y^2 = 4(x+4)^3 from 0 to 2

2. Find the arc length of the function: y = e^x from 0 to 1

Thank you very much for any assistance

**Soroban** · March 4th 2008, 08:41 AM

Hello, coolio!

The first one is straight-forward.
Exactly where is your difficulty?

1. Find the arc length of the function: $y^2 \:= \:4(x+4)^3$ from 0 to 2

Formula: . $L \;=\;\int^b_a\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\,dx$

We have: . $y \;=\;2(x+4)^{\frac{3}{2}}$

. .Then: . $\frac{dy}{dx} \:=\:3(x+4)^{\frac{1}{2}}$

. .And: . $1 +\left(\frac{dy}{dx}\right)^2 \;=\;1 + 9(x+4) \;=\;9x+37$

. . Hence: . $\sqrt{1+ \left(\frac{dy}{dx}\right)^2} \;=\;\sqrt{9x+37}$

Then we have: . $L \;=\;\int^2_0(9x+37)^{\frac{1}{2}}\,dx$

Can you finish it?

**coolio** · March 4th 2008, 09:06 AM

Thanks soroban, I tried multiplying out the polynomial and it just turned into a huge mess. I understand now

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