Problem 23 - Calculus of a Single Variable

Question

Finding Arc Length In Exercises \(17-26,\) (a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the are length. $$ x=e^{-y}, \quad 0 \leq y \leq 2 $$

Step-by-Step Solution

Answer

The arc length of the function \(x=e^{-y}\) over the interval from 0 to 2 can be approximated using the integral \(\int_{0}^{2}\sqrt{1+e^{-2y}}dy\). Since this integral is quite complex and cannot be evaluated with elementary methods, a graphing utility can be used to get an approximate value for the arc length.

Step 1: Sketching the graph

Using a graphing utility, plot the function \(x=e^{-y}\) on the interval \(0 \leq y \leq 2\). You will observe an exponential decay curve starting from \(x=1\) when \(y=0\) and approaching \(x=0\) when \(y=2\).

Step 2: Derivative of the function

Find the derivative of the function \(x=e^{-y}\). It gives us \(x'=-e^{-y}\).

Step 3: Setup the integral for the arc length

The arc length of a function over an interval [a, b] is given by \(\int_{a}^{b}\sqrt{1+(f'(x))^2}dx\). Here, \(f'(x)= -e^{-y}\). Thus, square the derivative and sum it with 1, we get \(1+(-e^{-y})^2\), which simplifies to \(1+e^{-2y}\). Hence, the integral becomes \(\int_{0}^{2}\sqrt{1+e^{-2y}}dy\).

Step 4: Approximating the arc length

To approximate the arc length, use a graphing utility to integrate \(\int_{0}^{2}\sqrt{1+e^{-2y}}dy\). The graphing utility will provide an approximate numerical value for the integral.

Key Concepts

Definite Integral

The concept of a definite integral is central to calculating arc length. A definite integral is used when you need to find the area under a curve between two specific points. It is represented as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits describing the interval over which you are integrating. In this particular exercise, you are asked to find the arc length of the curve described by \( x=e^{-y} \) over the interval \( 0 \leq y \leq 2 \). This means you will set up and use a definite integral to determine the precise length of that curve segment. However, it's important to note that not all integrals can be solved using simple techniques; some require more complex equations or numerical methods for approximation.

Derivative of Function

Understanding the derivative of a function is essential in calculating the arc length. The derivative gives us the slope of the tangent to the curve at any point along the curve. For the function \( x=e^{-y} \), the derivative is \( x'=-e^{-y} \).

To find the arc length of a curve, we incorporate the derivative into the arc length formula \( \int_{a}^{b}\sqrt{1+(f'(x))^2}dx \). This formula considers the rate at which the function's value changes as you move along the curve, and the derivative is the component that indicates this change. By plugging the derivative into the formula, you can correctly account for the slope's effect on the overall arc length.

Graphing Utility

A graphing utility is a computational tool used to visually represent functions and perform complex calculations like integration. In this exercise, using a graphing utility can help you plot the function \( x=e^{-y} \) over the specified interval \( 0 \leq y \leq 2 \).

While manual calculation of definite integrals can be daunting, graphing utilities simplify this task by providing numerical approximations. They can handle integrals that are difficult or impossible to solve analytically. By entering the integral \( \int_{0}^{2}\sqrt{1+e^{-2y}}dy \) into the graphing utility, it can compute an approximation for the arc length without needing to solve it manually.

Exponential Decay

Exponential decay is a process where a quantity decreases at a rate proportional to its current value. The function \( x=e^{-y} \) models this kind of behavior. Here, \( e \) is the base of the natural logarithm, which describes the continuous, decreasing nature of the curve.

As \( y \) increases, \( x \) approaches zero from one, illustrating this decay. Understanding this concept helps in visualizing and interpreting the curve's shape over the interval \( 0 \leq y \leq 2 \). Exponential decay occurs frequently in natural and financial processes, such as radioactive decay or depreciation, making it a useful concept across various domains.