Calculate length of the following function

General
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calculate length of the following function

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LectureNotes asked: “Calculate the length of the following function.” However, no specific function is provided. Could you clarify which function you’d like to evaluate?"

Answer:
Below is a general overview of how to calculate the length of a curve described by a function y = f(x) over an interval [a,b]:

  1. Formula for Arc Length
    The length L of the function y=f(x) from x=a to x=b is given by:

    L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx

  2. Clarify Variable Bounds

    • Ensure you know the interval [a,b] over which to integrate.
    • Compute the derivative \frac{dy}{dx} accurately.

  3. Set Up the Integral

    • Substitute \frac{dy}{dx} into the formula.
    • Integrate carefully, considering any simplifications or substitutions.

  4. Evaluate the Integral

    • Perform the integral step by step or use a tool if it’s not a standard integral.
    • Simplify the final result whenever possible.

Below is a summary table:

Step Description
1. Identify the function Ensure the function f(x) is clearly defined.
2. Determine the interval Specify the limits a and b for x.
3. Compute derivative Find \frac{dy}{dx} for f(x).
4. Apply arc length formula Use the formula: L = \int_a^b \sqrt{1 + (\frac{dy}{dx})^2} \, dx.
5. Solve the integral Integrate and simplify to find the exact or approximate length.

If you can provide the specific function—for instance, f(x) = x^2, f(x) = \sin x, or any other explicit form—then the steps above can be applied directly to calculate its arc length. Let me know, and I’ll help you with the detailed calculation.

@User