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Chapter 15: Problem 2

Find the arc length of \(f(x)=x^{2} / 8-\ln x\) on [1,2] .

Short Answer

Expert verified
The integral \(\int_{1}^{2} \sqrt{1+ x^{2} / 16 - 1 / 2 + 1 / x^{2}} dx\) could not be solved analytically. One can use numerical methods to find an approximate answer.

Step by step solution

01

Find derivative of \(f(x)\).

We first find the derivative of \(f(x) = x^{2} / 8 - \ln x\). Using the power rule and the derivative of the natural logarithm, we find \(f'(x) = x / 4 - 1 / x\).
02

Find \([f'(x)]^{2}\) and add 1.

We square the derivative we found and add 1. This gives us \(1 + [f'(x)]^{2} = 1 + [(x / 4 - 1 / x)]^{2}\) which simplifies to \(1 + x^{2} / 16 - 1 / 2 + 1 / x^{2}.\)
03

Compute the integral

We now compute the definite integral from 1 to 2 of the square root of the expression we derived in Step 2. \(\int_{1}^{2} \sqrt{1 + x^{2} / 16 - 1 / 2 + 1 / x^{2}} dx\). This integral is not easily solvable analytically and might require numerical methods for calculation.

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Key Concepts

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

Derivative Calculation
Understanding derivative calculation is essential for working with functions. A derivative tells us how steep a curve is at a given point by measuring the function's rate of change.

In this exercise, we have the function \( f(x) = \frac{x^2}{8} - \ln x \). To find the derivative, we apply basic differentiation rules:
  • The power rule helps us find the derivative of \( \frac{x^2}{8} \), giving us \( \frac{x}{4} \).
  • The derivative of \( \ln x \) is \( \frac{1}{x} \).
So, the derivative of \( f(x) \) is \( f'(x) = \frac{x}{4} - \frac{1}{x} \).

This derivative will allow us to progress to the next step in finding the arc length.
Definite Integral
Once we have the derivative, we need to use it to help find the arc length. The arc length formula involves integrating the function derived from the derivative.

In our case, after squaring the derivative \( f'(x) \), adding 1, and placing it inside a square root, we obtain \( \sqrt{1 + \left( \frac{x}{4} - \frac{1}{x} \right)^2} \).

The integral form to calculate arc length between \( x = 1 \) and \( x = 2 \) is given by:
  • \( \int_{1}^{2} \sqrt{1 + \frac{x^2}{16} - \frac{1}{2} + \frac{1}{x^2}} \, dx \)
This definite integral calculates a single value representing the exact arc length of the curve over the interval. It's a crucial tool that gives us detail about the segment of the curve itself.
Square Root Function
In searching for the arc length, the square root function plays a critical role.

The primary complexity in the definite integral \( \sqrt{1 + \left( \frac{x}{4} - \frac{1}{x} \right)^2} \) comes from the square root function.

However, a square root can often complicate the solution process when dealing with integrals. It adds a layer of complexity due to its nature of inverse power. Simplifying or estimating expressions containing a square root often requires creativity or approximation methods.

In practical terms, understanding how to manage and compute square roots, especially in integrals, is an essential skill in calculus.
Numerical Methods
Sometimes, definite integrals like the one for arc length cannot be solved using standard analytical methods.

Numerical methods provide an alternative approach to solutions that can't be easily found otherwise. These methods use algorithms to approximate the value of an integral.
  • Methods such as trapezoidal rule or Simpson鈥檚 rule can offer approximate values for integrals.
  • These techniques are not exact but can provide sufficiently precise answers for most practical purposes.
In our scenario, using numerical methods allows us to handle integrals involved in finding the arc length, which may not simplify neatly otherwise. Often, numerical solutions are key when dealing with real-world data where exact answers are either impossible or impractical to obtain analytically.

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Most popular questions from this chapter

A hemispheric bowl of radius \(r\) contains water to a depth \(h\). Find the volume of water in the bowl.

Let \(S\) be the region of the \(x y\) -plane bounded above by the curve \(x^{3} y=64,\) below by the line \(y=1,\) on the left by the line \(x=2,\) and on the right by the line \(x=4 .\) Find the volume of the solid obtained by rotating \(S\) around (a) the \(x\) -axis, (b) the line \(y=1,\) (c) the \(y\) -axis, (d) the line \(x=2\).

Use integration to find the volume of the solid obtained by revolving the region bounded by \(x+y=2\) and the \(x\) and \(y\) axes around the \(x\) -axis.

Use integration to compute the volume of a sphere of radius \(r\).

The equation \(x^{2} / 9+y^{2} / 4=1\) describes an ellipse. Find the volume of the solid obtained by rotating the ellipse around the \(x\) -axis and also around the \(y\) -axis. These solids are called ellipsoids; one is vaguely rugby-ball shaped, one is sort of flying-saucer shaped, or perhaps squished- beach-ball-shaped.
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