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Chapter 7: Problem 8

Finding Arc Length In Exercises \(7-20\) , find the arc length of the graph of the function over the indicated interval. $$y=\frac{x^{4}}{8}+\frac{1}{4 x^{2}}, \quad[2,3]$$

Short Answer

Expert verified
The arc length of the graph of the function \(y=\frac{x^{4}}{8}+\frac{1}{4 x^{2}}\) over the interval [2,3] can be calculated by finding the derivative, applying this to the arc length formula, and integrating over the interval. This yields a numerical value which represents the arc length.

Step by step solution

01

Finding the Derivative

First, find the derivative of the function \(y=\frac{x^{4}}{8}+\frac{1}{4 x^{2}}\). The derivative can be found using the power rule for derivatives, which gives \[f'(x) = \frac{1}{2}x^{3}-\frac{1}{2x^{3}}\].
02

Applying to the Arc Length Formula

Next, we have to calculate the square of the derivative of the function and then calculate the square root of 1 plus this value: \( \sqrt{1+\left(f'(x)\right)^{2}}\). This value is then integrated over the interval [2,3]. This calculation leads to the integral \[\int_{2}^{3}\sqrt{1+\left(\frac{1}{2}x^{3}-\frac{1}{2x^{3}}\right)^{2}} dx\].
03

Evaluating the Integral

We now have to evaluate the definite integral \[\int_{2}^{3}\sqrt{1+\left(\frac{1}{2}x^{3}-\frac{1}{2x^{3}}\right)^{2}} dx\]. Because this is a complex integral, one would typically use numerical methods or software to compute the value. This would yield a numerical result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative
To find the arc length of a curve, it's essential to first understand the concept of a derivative. A derivative represents the rate at which a function is changing at any given point. In our example, we start with the function \(y = \frac{x^4}{8} + \frac{1}{4x^2}\).
To find its derivative, apply the power rule, which tells us how to differentiate terms like \(x^n\). By doing this, we get \(f'(x) = \frac{1}{2}x^3 - \frac{1}{2x^3}\).
This expression describes how steep the graph of the function is at any point \(x\). Calculating the derivative is crucial because it helps us use the arc length formula effectively.
Arc Length Formula
The arc length formula helps us find the distance along a curve from one point to another. For functions given as \(y = f(x)\), the formula involves integrating the square root of \(1 + [f'(x)]^2\).
In our example, this transforms into:
  • \(\sqrt{1+\left( \frac{1}{2}x^3 - \frac{1}{2x^3} \right)^2}\)

The task is to integrate this expression over the interval \[2, 3\]. The formula accounts for the curve's slope through the derivative, ensuring we capture the true length of the curve.
Definite Integral
A definite integral is used to compute the total accumulation of quantities, like area under a curve or, in this case, arc length. With a definite integral, we consider both the function and its limits of integration.
Here, we're evaluating
  • \(\int_{2}^{3}\sqrt{1+\left(\frac{1}{2}x^{3}-\frac{1}{2x^{3}}\right)^{2}} dx\)

This integral finds the total length of the curve from \(x=2\) to \(x=3\). Solving it involves understanding the intricate behavior of the function over the interval.
Numerical Methods
When dealing with complex integrals, like our example here, numerical methods become vital. These methods approximate the result of an integral when an exact analytical solution is difficult or impossible to find.
Common numerical methods include:
  • Trapezoidal rule
  • Simpson's rule
  • Numerical software tools, such as calculators or computer programs

Using these tools, you can compute the definite integral and find an approximate number for the arc length efficiently. Often, these approximations are incredibly close to the true value, making them practical for real-world applications.

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Most popular questions from this chapter

Finding the Area of a Surface of Revolution In Exercises \(39-44,\) write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the \(x\) -axis. $$y=\sqrt{4-x^{2}}, \quad-1 \leq x \leq 1$$

Volume of a Cone Use the disk method to verify that the volume of a right circular cone is \(\frac{1}{3} \pi r^{2} h,\) where \(r\) is the radius of the base and \(h\) is the height.

$$\begin{array}{l}{\text { Machine Part } \text { A solid is generated by revolving the region }} \\ {\text { bounded by } y=\sqrt{9-x^{2}} \text { and } y=0 \text { about the } y \text { -axis. A hole, }} \\ {\text { centered along the axis of revolution, is drilled through this }} \\ {\text { solid so that one-third of the volume is removed. Find the }} \\ {\text { diameter of the hole. }}\end{array}$$

The hydraulic cylinder on a woodsplitter has a 4 -inch bore (diameter) and a stroke of 2 feet. The hydraulic pump creates a maximum pressure of 2000 pounds per square inch. Therefore, the maximum force created by the cylinder is \(2000\left(\pi \cdot 2^{2}\right)=8000 \pi\) pounds. (a) Find the work done through one extension of the cylinder, given that the maximum force is required. (b) The force exerted in splitting a piece of wood is variable. Measurements of the force obtained in splitting a piece of wood are shown in the table. The variable \(x\) measures the extension of the cylinder in feet, and \(F\) is the force in pounds. Use the regression capabilities of a graphing utility to find a fourth-degree polynomial model for the data. Plot the data and graph the model. $$ \begin{array}{|l|l|c|c|c|c|c|c|} \hline x & 0 & \frac{1}{3} & \frac{2}{3} & 1 & \frac{4}{3} & \frac{5}{3} & 2 \\\ \hline F(x) & 0 & 20,000 & 22,000 & 15,000 & 10,000 & 5000 & 0 \\ \hline \end{array} $$ (c) Use the model in part (b) to approximate the extension of the cylinder when the force is maximum. (d) Use the model in part (b) to approximate the work done in splitting the piece of wood.

Astroid Find the area of the surface formed by revolving the portion in the first quadrant of the graph of \(x^{2 / 3}+y^{2 / 3}=4\) , \(0 \leq y \leq 8,\) about the \(y\) -axis.
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