Chapter 6: Q. 23 (page 539)

Approximate the arc length of f (x) on [a, b], using n line segments and the distance formula. Include a sketch of f (x) and the line segments .
$f (x) = \sqrt{9 - x^{2}}, [a, b] = [- 3, 3], n = 3$

Short Answer

Expert verified

The arc length is $13.996$ .

The graph is as below:

Step by step solution

Step 1. Given information .

Consider the given function $f (x) = \sqrt{9 - x^{2}}$ .

Step 2. Formula used .

Arc length $= \lim_{n \to \infty} \sum_{k = 1}^{n} \sqrt{1 + {(\frac{\partial y_{k}}{\partial x})}^{2}} \cdot \partial x$

Step 3. Find the arc length .

$f (x) = \sqrt{9 - x^{2}}, a = - 3, b = 3, n = 3 \partial x = \frac{b - a}{n} = \frac{3 + 3}{3} = 2 x_{k} = a + k \partial x x_{0} = - 3 + 0 = - 3, x_{1} = - 3 + 1 \times 2 = - 1 x_{2} = - 3 + 2 \times 2 = 1, x_{3} = - 3 + 3 \times 2 = 3 \partial y_{1} = f (x_{1}) - f (x_{0}) = \sqrt{8} - 0 = \sqrt{8} = 2 \cdot 828 \partial y_{2} = f (x_{2}) - f (x_{1}) = \sqrt{8} - \sqrt{8} = 0 \partial y_{3} = f (x_{3}) - f (x_{2}) = 0 - \sqrt{8} = - 2.828 A r c l e n g t h = [\sqrt{1 + {(\frac{2.828}{2})}^{2}} + \sqrt{1 + 0} + \sqrt{1 + {(\frac{- 2.828}{2})}^{2}}] \times 2 = [2 \sqrt{1 + {(1.414)}^{2}} + 1] \times 2 = (2 \times 2.999 + 1) \times 2 = 13.996$

Step 4. Sketch of the function fx=9-x2.

The sketch of the given function is shown below .

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Approximate the arc length of f (x) on [a, b], using n line segments and the distance formula. Include a sketch of f (x) and the line segments .
$f (x) = \sqrt{9 - x^{2}}, [a, b] = [- 3, 3], n = 3$

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the arc length .

Step 4. Sketch of the function fx=9-x2.

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Approximate the arc length of f (x) on [a, b], using n line segments and the distance formula. Include a sketch of f (x) and the line segments .fx=9-x2,a,b=-3,3,n=3

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the arc length .

Step 4. Sketch of the function fx=9-x2.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Pure Maths

Logic and Functions

Mechanics Maths

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Approximate the arc length of f (x) on [a, b], using n line segments and the distance formula. Include a sketch of f (x) and the line segments .
$f (x) = \sqrt{9 - x^{2}}, [a, b] = [- 3, 3], n = 3$