91Ó°ÊÓ

Calculating derivatives is an important skill in calculus, particularly when working with functions to find rates of change. In this exercise, we begin by finding the derivative of the given function:
\[ y = \frac{1}{8} x^{4} + \frac{1}{4 x^{2}} \]
To differentiate this, apply basic derivative rules:

• For \( x^n \), the derivative is \( nx^{n-1} \), so \( \frac{1}{8} x^{4} \) becomes \( \frac{1}{2} x^{3} \).
• For \( x^{-n} \), the derivative is \( -nx^{-n-1} \), so \( \frac{1}{4 x^{2}} \) becomes \( -\frac{1}{2 x^{3}} \).

Putting these together, the derivative is:\[ y' = \frac{1}{2} x^{3} - \frac{1}{2 x^{3}} \]This expression will be used in subsequent steps, particularly when calculating the arc length. Understanding how to derive it ensures you can handle similar problems.

Integral calculus is crucial for finding areas under curves and, in this case, calculating arc length. Once you've computed the derivative of a function, squaring it forms part of the arc length calculation:
\[ (y')^{2} = \left(\frac{1}{2} x^{3} - \frac{1}{2 x^{3}} \right)^{2} \]
Next, you simplify and prepare it for integration by adding 1:

• First, expand the squared term: \( \frac{1}{4} x^{6} - \frac{1}{2} + \frac{1}{4 x^{6}} \)
• Then, add 1 to this expression: \( \frac{1}{4} x^{6} + \frac{1}{2} + \frac{1}{4 x^{6}} \).

Using integral calculus, we express this as an integral over the interval \([-2, -1]\):\[ L = \int_{-2}^{-1} \sqrt{\frac{1}{4} x^{6} + \frac{1}{2} + \frac{1}{4 x^{6}}} \, dx \]This integral will give us the exact arc length once calculated, highlighting integral calculus's power in practical applications.

The formula for arc length is fundamental in determining the length of a curve over a particular interval. In this example, after deriving and simplifying the equation,
we form our integral as:\[ L = \int_{-2}^{-1} \sqrt{1+(y')^{2}} \, dx \]
where \( (y')^{2} = \frac{1}{4} x^{6} + \frac{1}{2} + \frac{1}{4 x^{6}} \).
Integrating this expression offers insights into the distance along the curve from \( x = -2 \) to \( x = -1 \).

To effectively compute this, one might utilize numerical integration or a computer algebra system, especially when the expression is complex. The arc length formula is especially useful in physics and engineering, where it often aids in the design and analysis of curved structures.