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

Step 1. Given information .

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

Step 2. Formula used .

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

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

Step 3. Find the arc length .

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

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

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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

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the arc length .

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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 .fx=x32,a,b=0,2

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the arc length .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

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Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

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 .
$f (x) = x^{\frac{3}{2}}$ , $[a, b] = [0, 2]$