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Chapter 6: Q. 49 (page 499)

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.
${鈭�}_{2}^{3} \sqrt{\frac{x^{4} + 1}{x^{4}}} d x$

Short Answer

Expert verified

The function is $f (x) = \frac{1}{x}$ and $[a, b] = [2, 3]$ .

Step by step solution

01

Step 1. Given information.

Consider the given integral is ${鈭�}_{2}^{3} \sqrt{\frac{x^{4} + 1}{x^{4}}} d x$ .

02

Step 2. Use arc length formula.

The formula for a function to find the arc length from $x = a$ to $x = b$ is given by ${鈭�}_{a}^{b} \sqrt{1 + {(f^{'} (x))}^{2}} d x$ .

03

Step 3. Find derivative of given function.

Compare given definite integral function with the arc length formula.

$\begin{array}{rcl} {鈭�}_{a}^{b} \sqrt{1 + {(f^{'} (x))}^{2}} d x & = & {鈭�}_{2}^{3} \sqrt{\frac{x^{4} + 1}{x^{4}}} d x \\ = & {鈭�}_{2}^{3} \sqrt{1 + {(- \frac{1}{x^{2}})}^{2}} d x \end{array}$

Therefore the derivative function is $f^{'} (x) = (- \frac{1}{x^{2}})$ and limit is $[a, b] = [2, 3]$ .

04

Step 4. Find function fx.

The function whose differentiation is $(- \frac{1}{x^{2}})$ is $f (x) = \frac{1}{x}$ .

Thus the required function is $f (x) = \frac{1}{x}$ and $[a, b] = [2, 3]$ .

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Most popular questions from this chapter

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52.

35. $\frac{d y}{d x} = \frac{3}{y}, y (0) = 2$

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.

$\frac{d y}{d x} = \frac{x + 1}{\sqrt{x}}$

find an equation that gives y as an implicit function of x. Then draw the continuous curve that satisfies this differential equation and passes through the point (2, 0).

$\frac{d y}{d x} = \frac{鈭� x}{y}$

Consider the region between $f (x) = \sqrt{x}$ and the x-axis on $[0, 4]$ . For each line of rotation given in Exercises 27鈥�30, use four disks or washers based on the given rectangles to approximate the volume of the resulting solid.

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.

$\frac{d y}{d x} = \frac{3 x + 1}{x y}$

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