# How do you find the arc length of the curve #y = (x^4/8) + (1/4x^2) # from [1, 2]?

If you don't remember the arc length formula, you can use the distance formula:

So, first let's take the derivative and square it.

Now you can insert this as:

Now I want to somehow get this into a perfect square under a square root if possible. This was:

...and as it turns out, there is no answer in terms of standard mathematical functions, so we have to stop writing here.

All you can do is numerically evaluate this on your calculator or Wolfram Alpha, so if it weren't for the boundaries, this would not be that fun!

graph{(y - x^4/8 - x^2/4)*sqrt(1^2 - (x-1)^2)/sqrt(0.5^2 - (x-1.5)^2) = 0 [-2.92, 5.85, -0.955, 3.428]}

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To find the arc length of the curve y = (x^4/8) + (1/4x^2) from x = 1 to x = 2, you use the formula for arc length:

L = ∫√(1 + (dy/dx)^2) dx

First, find dy/dx:

dy/dx = (x^3/2) - (1/2x^3)

Next, square it:

(dy/dx)^2 = (x^6/4) - (x^2) + (1/4x^6)

Now, plug into the arc length formula and integrate from 1 to 2:

L = ∫√(1 + (x^6/4) - (x^2) + (1/4x^6)) dx from 1 to 2

After integrating, you'll find the arc length.

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