Finding the Arc Length of a Function over a Specified Interval

Introduction

In mathematics, the arc length of a function is the length of the curve represented by the function over a specific interval. Calculating the arc length of a function can be a useful tool in various applications, such as engineering, physics, and geometry. In this article, we will learn how to find the arc length of the graph of the function y= 1/2(e^x+e^-x) over the interval [0,2].

Step 1: Understand the Function

The function we are working with is y= 1/2(e^x+e^-x), where e is the base of the natural logarithm. This function is a hyperbolic cosine function, which is a commonly used function in mathematics and physics.

Step 2: Derive the Formula for Arc Length

The formula for the arc length of a function y=f(x) over the interval [a,b] is:

Arc Length = ∫(a to b) √(1 + (df/dx)^2) dx

In our case, the interval is [0,2], so we need to find the derivative of the function and then integrate the expression over the interval.

Step 3: Find the Derivative of the Function

The derivative of the function y= 1/2(e^x+e^-x) is:

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

Step 4: Substitute the Derivative into the Arc Length Formula

Substituting the derivative into the arc length formula, we get:

Arc Length = ∫(0 to 2) √(1 + (1/2(e^x - e^-x))^2) dx

Step 5: Evaluate the Integral

Evaluating the integral, we get:

Arc Length = 2.718 (rounded to three decimal places)

Conclusion

In this article, we have learned how to find the arc length of the graph of the function y= 1/2(e^x+e^-x) over the interval [0,2]. We derived the formula for arc length, found the derivative of the function, and then evaluated the integral to obtain the final result. This process is a common task in high school mathematics and can be applied to a variety of functions and intervals.