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Chapter 6: Q. 51 (page 499)

Each definite integral represents the arc length of a function f(x) on an interval [a, b]. Determine the function and interval.
${鈭�}_{0}^{蟺 / 4} s e c x d x$

Short Answer

Expert verified

The function is $f (x) = \ln |\cos x|$ and interval is $[a, b] = [0, \frac{蟺}{4}]$ .

Step by step solution

01

Step 1. Given information.

Consider the given function is ${鈭�}_{0}^{蟺 / 4} s e c x d x$ .

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 byrole="math" localid="1649290930947" ${鈭�}_{a}^{b} \sqrt{1 + {(f^{'} (x))}^{2}} d x$ .

03

Step 3. Find derivative of given function.

Compare given definite integral function with the arc length formula.

$\begin{array}{rcl} {鈭�}_{a}^{b} \sqrt{1 + {(f^{'} (x))}^{2}} d x & = & {鈭�}_{0}^{蟺 / 4} s e c x d x \\ = & {鈭�}_{0}^{蟺 / 4} \sqrt{1 + {(\tan x)}^{2}} d x \end{array}$

Therefore the function will be $f^{'} (x) = \begin{array}{rcl} \tan \end{array} \begin{array}{rcl} x \end{array}$ and interval isrole="math" localid="1649291054388" $[a, b] = [0, \frac{蟺}{4}]$ .

04

Step 4. Find function fx.

The function whose differentiation is $\tan x$ is $\ln |\cos x|$ .

Thus the required function is $f (x) = \ln |\cos x|$ and $[a, b] = [0, \frac{蟺}{4}]$ .

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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.

$\frac{d y}{d x} = y \sin 鈦� x$

In the process of solving $\frac{d y}{d x} = y$ by separation of variables, we obtain the equation $| y | = e^{x + C}$ . After solving for $y$ , this equation becomes $y = A e^{x}$ . How is $A$ related to $C$ ? What happened to the absolute value?

For each pair of definite integrals in exercise 13-18 decide which if either is larger without computing the integrals

$蟺 {鈭�}_{0}^{3} x^{2} d x a n d 蟺 {鈭�}_{0}^{3} (9 - x^{2}) d x$

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

36. $\frac{d y}{d x} = \frac{6}{x}, y (1) = 7$

See all solutions

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