Chapter 6: Q. 27 (page 539)

Approximate the arc length of f (x) on [a, b], using the approximation $\sum_{k = 1}^{n} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$ with the given value of n. In each problem list the values of $δ y_{k}$ for role="math" localid="1649228183425" $k = 0, 1, 2, 3 . . . . . n$ .
$f (x) = I n x, [a, b] = [1, 3], n = 4$

Short Answer

Expert verified

The arc length is $\frac{\sqrt{13} + 3 \sqrt{5}}{4}$ .

Step by step solution

Step 1. Given information .

Consider the given function $f (x) = I n x, [a, b] = [1, 3], n = 4$ .

Step 2. Formula used .

Arc length of $f (x) = \sum_{k = 1}^{n} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$ .

Step 3. Find the arc length .

$f (x) = \sum_{k = 1}^{n} \sqrt{1 + {(\frac{δ y_{k}}{δ x})}^{2}} \cdot δ x$

$δ x = \frac{b - a}{n} = \frac{1}{2} x_{0} = a + k \cdot δ x = 1, x_{1} = \frac{3}{2}, x_{2} = 2 x_{3} = \frac{5}{2}, x_{4} = 3 δ y_{1} = f (x_{1}) - f (x_{0}) = \frac{3}{2} δ y_{2} = f (x_{2}) - f (x_{1}) = \frac{1}{2} δ y_{3} = f (x_{3}) - f (x_{2}) = \frac{1}{2} δ y_{4} = f (x_{4}) - f (x_{3}) = \frac{1}{2}$

$A r c l e n g t h = \sqrt{1 + {(\frac{δ y_{1}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{2}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{3}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{4}}{δ x})}^{2}} \cdot δ x = \frac{\sqrt{13}}{4} + \frac{\sqrt{5}}{4} + \frac{\sqrt{5}}{4} + \frac{\sqrt{5}}{4} = \frac{\sqrt{13} + 3 \sqrt{5}}{4}$

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Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find the arc length .

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Approximate the arc length of f (x) on [a, b], using the approximation ∑k=1n1+δykδx2·δxwith the given value of n. In each problem list the values of δykfor role="math" localid="1649228183425" k=0,1,2,3.....n.fx=Inx,a,b=1,3,n=4

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

Pure Maths

Calculus

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.