Find the arc length given the equation y=(x^4/8)+(1/4x^2) [1,3]
Do you mean between x = 1 abd x = 3?
Is the second term 1/(4x^2) or (1/4)x^2?
Yes between the interval x=1 and x=3 and in the second term it is 1/(4x^2)
To find the arc length of a curve given an equation, we can use the formula for arc length:
\(L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx\)
Here, \(a\) and \(b\) represent the range of values of \(x\) on which our curve is defined. In this case, the given range is \([1, 3]\).
First, let's find \( \frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\):
\(y = \frac{x^4}{8} + \frac{1}{4x^2}\)
\(\frac{dy}{dx} = \frac{1}{8} \cdot 4x^3 - \frac{1}{4} \cdot 2x^{-3}\)
\(\frac{dy}{dx} = \frac{x^3}{2} - \frac{1}{2x^3}\)
Next, insert this derivative into the formula for arc length:
\(L = \int_{1}^{3} \sqrt{1 + \left(\frac{x^3}{2} - \frac{1}{2x^3}\right)^2} dx\)
Now, we need to simplify this equation before we can integrate it.