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Chapter 6: Q. 34 (page 539)

Find the exact value of the arc length of each function f (x) on [a, b] by writing the arc length as a definite integral and then solving that integral .
$f (x) = x^{\frac{3}{2}}$ , $[a, b] = [0, 2]$

Short Answer

Expert verified

The arc length is $\frac{13}{4}$ .

Step by step solution

01

Step 1. Given information .

Consider the given function $f (x) = x^{\frac{3}{2}}$ .

02

Step 2. Formula used .

The formula used to find the arc length from $x = a$ to $x = b$ is ,

Arc length of $f (x) = {鈭�}_{a}^{b} \sqrt{1 + {(f' (x))}^{2}} d x$ .

03

Step 3. Find the arc length .

$f (x) = x^{\frac{3}{2}} f' (x) = \frac{d}{d x} [f (x)] = \frac{3}{2} x^{\frac{1}{2}}$

$A r c l e n g t h = {鈭�}_{0}^{2} \sqrt{1 + {(\frac{3}{2} x^{\frac{1}{2}})}^{2}} d x = {鈭�}_{0}^{2} \sqrt{1 + \frac{9}{4} x} d x = {鈭�}_{0}^{2} {(1 + \frac{9}{4} x)}^{\frac{1}{2}} d x = \frac{1}{2} {鈭�}_{0}^{2} (1 + \frac{9}{4} x) d x = \frac{1}{2} {[x + \frac{9 x^{2}}{8}]}_{0}^{2} = \frac{1}{2} [2 + \frac{9 路 4}{8}] = \frac{13}{4}$

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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 29-52.

$\frac{d y}{d x} = y^{2} \cos x, y (蟺) = 1$

Each of the definite integrals in Exercises 19鈥�24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.

$蟺 {鈭�}_{0}^{4} (2^{2} - {(\sqrt{y})}^{2}) dy$

Consider the region between the graph of $f (x) = \frac{3}{x}$ and the x-axis on [1,3]. For each line of rotation given in Exercises 31鈥� 34, use definite integrals to find the volume of the resulting solid.

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

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

Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f (x) = \sqrt{1 - x^{2}}$ , $[a, b] = [- 1, 1]$

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