Find the arc length of y=((x)/(2))^4+(1)/(2 x^2) over [1,4] . Hint: Show that 1+(y^')^2 is a per

step 1 Find the arc length of the given curve over the given interval. We have the curve y of x equal t

Find the arc length of y=((x)/(2))^4+(1)/(2 x^2) over [1,4]...

Key Concepts

Arc Length Formula

The arc length of a curve defined by a function is calculated using the formula L = ?[a to b] ?(1 + (dy/dx)²) dx. This formula establishes the relationship between the derivative of the function and the length of the curve over a specified interval, and it is a fundamental tool in calculus for measuring distances along curves.

Differentiation

Differentiation involves computing the derivative, dy/dx, of the given function. The derivative represents the instantaneous rate of change of the function with respect to x and is essential in determining the integrand for the arc length formula. Techniques such as the power rule and chain rule are typically used to calculate the derivative of functions involving polynomial or rational expressions.

Algebraic Simplification and Perfect Squares

Once the derivative dy/dx is computed, the next step involves simplifying the expression inside the square root of the arc length formula, which is 1 + (dy/dx)². Recognizing that this expression can be arranged into a perfect square allows one to replace the square root of a square with a simpler algebraic expression. This process of algebraic manipulation simplifies the integration process by reducing a more complicated integrand into a form that is easier to integrate.

Definite Integration

After simplifying the integrand, the next step requires evaluating the definite integral over the given interval [a, b]. This involves applying integration techniques to compute the total arc length. The proper setup of the integral and the evaluation of its bounds are critical steps in arriving at the final solution for the arc length of the curve.

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