Question

Approximation In Exercises 25 and \(26,\) approximate the arc length of the graph of the function over the interval [0,4] in four ways. (a) Use the Distance Formula to find the distance between the endpoints of the arc. (b) Use the Distance Formula to find the lengths of the four line segments connecting the points on the arc when \(x=0, x=1, x=2, x=3,\) and \(x=4\). Find the sum of the four lengths. (c) Use Simpson's Rule with \(n=10\) to approximate the integral yielding the indicated arc length. (d) Use the integration capabilities of a graphing utility to approximate the integral yielding the indicated arc length. $$ f(x)=\left(x^{2}-4\right)^{2} $$

Short answer

The short answer will come in four different parts, each one corresponding to the approximation made in each step. A numerical answer can be obtained for each of these steps, but the exact value will depend on the specific calculations made and the precision of the methods used.

Step-by-step solution

Distance Formula between Endpoints

The endpoints of the arc over the interval [0,4] are \(f(0)\) and \(f(4)\). Applying the Distance Formula we would calculate it as the difference between these two points: \(|f(4) - f(0)|\). Calculate this to get the approximate arc length for method (a).

Distance Formula for Line Segments

For method (b), calculate the lengths of the four line segments connecting the points on the arc at \(x=0, x=1, x=2, x=3,\) and \(x=4\). Using the distance formula, calculate the distances between these points and sum them up: \(|f(1)-f(0)| + |f(2)-f(1)| + |f(3)-f(2)| + |f(4)-f(3)|\).

Simpson’s Rule Approximation

To approximate the arc length using Simpson's Rule with \(n=10\) as in method (c), start by determining the expression for arc length, which is given by \(\int_{a}^{b}\sqrt{1+(f'(x))^2}{dx}\). Here, \(f'(x) = 4x(x^{2} - 4)\). Compute the definite integral using Simpson’s Rule for an approximate arc length.

Graphing Utility Approximation

The final approximation is performed using the integration capabilities of a graphing utility as described in method (d). This will require inputting the expression for arc length into the utility and computing the definite integral over the required interval. Although this method is largely dependent on the capabilities of the utility used, it's essentially a matter of inputting \(\int_{0}^{4}\sqrt{1+(4x(x^{2} - 4))^{2}}{dx}\) and interpreting the output.

Key concepts

Distance Formula

The Distance Formula is a foundational concept in geometry, crucial for understanding the arc length approximation exercise. It is derived from the Pythagorean theorem and allows us to calculate the distance between two points in a coordinate system. The Distance Formula is defined as:
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Where \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of the two points. In the context of the exercise, students must first understand that an arc on a graph can be approached as a series of straight line segments between points. This approximation gets better as more segments are used.

When calculating arc length, the Distance Formula is used to measure the distance between consecutive points along the function's curve. For a closer approximation, the function's curve is divided into smaller segments, and distances between these points are found using the formula and then summed. This approach, while not as accurate as using calculus methods, provides a simpler alternative for estimating lengths of curves.

Simpson's Rule

Simpson's Rule is a numerical method for estimating the value of a definite integral, which is particularly useful when an exact antiderivative is difficult to find. It's a very important tool in calculus for approximating areas under curves and is defined by the formula:
\( S_n = \frac{\Delta x}{3} [y_0 + 4(y_1 + y_3 + ... + y_{n-1}) + 2(y_2 + y_4 + ... + y_{n-2}) + y_n] \)
Here, \( y_i \) are the function values at equally spaced points, \( \Delta x \) is the width of each subinterval, and \( n \) is an even number of intervals.

In the given exercise, the integral of the square root of \( 1+(f'(x))^2 \) determines the precise arc length. Using Simpson's Rule, students employ an approximation method that involves sampling the function at regular intervals and applying the formula. For the best results, a higher number of intervals (n) can be chosen, which provides a more precise approximation. This rule gives a very close approximation to the true value of the integral when the function is reasonably smooth.

Definite Integral

The definite integral is at the heart of calculus and represents the accumulation of quantities, such as area under a curve, between two points. Mathematically, it is expressed as:
\( \int_{a}^{b} f(x) \,dx \)
Where \( a \) and \( b \) are the lower and upper bounds, respectively, and \( f(x) \) is the function being integrated. A fundamental application of the definite integral is calculating the arc length of a function over a certain interval.

To understand the exercise on arc length approximation, students need to grasp the concept that the arc length of a smooth, continuous function from \( a \) to \( b \) can be found using a definite integral. The arc length formula incorporates the derivative of the function because it involves the function's rate of change, which relates to the steepness or slope at any point along the curve. The marriage of geometry and calculus through the definite integral enables precise computations of lengths of curves that are otherwise complex or impossible to measure directly.

Graphing Utility

A graphing utility is a powerful tool for visualizing and calculating properties of functions, often used in math education and professional practices. When it comes to arc length approximations, a graphing utility can quickly compute a definite integral, especially for complex functions where manual calculations are cumbersome.

In our textbook exercise, the graphing utility's integration function comes into play for method (d). Students are to enter the definite integral of the square root of \( 1+(f'(x))^2 \) over the given interval to obtain the arc length. This immersive, technological approach does not only yield a swift result but also helps in visualizing the problem. It gives a real-time dynamic graph of the function which assists in understanding the nature of the curve whose arc length is being approximated. Although this method requires less manual computation, it is imperative that students understand the underlying concepts, such as definite integrals, to validate the results provided by the graphing utility.