Problem 25 - 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. $$ y=2 \arctan x, \quad 0 \leq x \leq 1 $$

Step-by-Step Solution

Answer

The definite integral that represents the arc length of the curve from \(0 \leq x \leq 1\) is \( \int_{0}^{1} \sqrt{1 + \left( \frac{2}{1 + x^2} \right)^2} dx\). This integral can't be easily evaluated using basic integration techniques and needs the capabilities of a graphing utility or numeric integration techniques for a solution.

Step 1: Sketching the Graph

First step is to sketch the graph of the function \(y = 2 \arctan(x)\). You would find that the graph starts from the origin and rises before it gradually flattens as it approaches \(x = 1\). The area of interest will be highlighted (i.e., the area between \(0 \leq x \leq 1\)).

Step 2: Integrate the Function for Arc Length

The formula for the arc length of a function \(f\) from \(a\) to \(b\) is given by \( \int_{a}^{b} \sqrt{1 + (f'(x))^2} dx\). So, first we need to find the derivative of the function \(y = 2 \arctan(x)\). The derivative \(f'(x)\) is \(2/(1+x^2)\). Then, plug \(f'(x)\) into the arc length formula and simplify as much as possible to get the definite integral for the arc length, which is \( \int_{0}^{1} \sqrt{1 + \left( \frac{2}{1 + x^2} \right)^2} dx\).

Step 3: Observe the difficulties in evaluating the Integral and using alternate methods

Observe that the obtained integral is not easily solvable by basic integral techniques. Therefore, to solve this integral, utilities of a graphing calculator or other numeric integration techniques can be used. This is the integral you would type into your calculator, or whatever software/method you are using. For this exercise, we'll not solve the integral, but in practice you would at this point attempt to find a numerical approximation.

Key Concepts

Definite Integral

The definite integral is a concept at the heart of calculus, primarily concerned with the accumulation of quantities and the computation of areas under curves. When dealing with the arc length of a curve, the definite integral facilitates the quantification of this length over a specific interval.

Imagine plotting the curve of a function on a graph and then measuring the distance along the curve between two points on the x-axis; this is what calculating the arc length is about. Mathematically, the arc length from point 'a' to point 'b' on the function is represented by the integral of the square root of 1 plus the derivative of the function squared, integrated over the interval [a, b].

Applying this to the given exercise, where the function is the arctangent of x, multiplied by 2, we would seek to compute the integral from 0 to 1. The square root in the arc length formula adds complexity, creating a scenario where evaluating the integral directly may be challenging or impossible using elementary antiderivative techniques, leading us to the alternative method of numerical integration for approximation.

Numerical Integration

In many cases, a definite integral cannot be solved exactly using elementary methods, leading us to turn to numerical integration. Numerical integration is a collection of algorithms and procedures designed to approximate the value of a definite integral. It's akin to estimating the area under a curve by adding up areas of simple geometric shapes that approximate the curve.

There are various methods of numerical integration, such as the Trapezoidal Rule, Simpson's Rule, or Monte Carlo methods. Each technique has its own balance of complexity, accuracy, and computational intensity, but all serve the same purpose: to provide an estimated value for an integral that isn't readily solvable.

In our arctangent example, after realizing that we cannot integrate the arc length formula by hand, we would use a graphing utility or a software tool that employs one of these numerical methods to find an approximation of the arc length for the interval from 0 to 1.

Derivative of a Function

The derivative of a function represents the rate at which the function's output changes with respect to its input. In other words, it's the slope of the function's curve at any given point. For computing arc lengths, the derivative gives us an essential part of the formula by showing how the function 'moves' at every point along the x-axis.

The derivative provides information about the curvature and steepness of the graph, which is crucial for determining how 'long' the curve is. In the context of our exercise with the function (\(y = 2 \times \text{arctan}(x)\)), we calculate the derivative to be (\(2/(1+x^2)\)). It's this expression, squared, that we add to 1 under the square root in the arc length integral.

This process highlights an important connection in calculus between derivatives and integrals, specifically how the rate of change at every point (given by the derivative) affects the overall accumulation (the integral) across an interval, like calculating the length of a curve.