Chapter 6: Q. 26 (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 $y_{k} = 0, 1, 2, 3 . . . . . . . . n$ .
$f (x) = \cos x, [a, b] = [0, π], n = 3$

Short Answer

Expert verified

The arc length is $2 \sqrt{3601} + \sqrt{14401}$ .

Step by step solution

Step 1. Given information .

Consider the given function $f (x) = \cos x$ .

Step 2. Formula used .

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

Step 3. Find arc length .

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

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

The arc length of $f (x) = \lim_{n \to \infty} \sum_{k = 1}^{3} \sqrt{1 + {(\frac{δ y_{1}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{2}}{δ x})}^{2}} \cdot δ x + \sqrt{1 + {(\frac{δ y_{3}}{δ x})}^{2}} \cdot δ x = \sqrt{1 + {(\frac{- 1}{60})}^{2}} \cdot 60 + \sqrt{1 + {(\frac{- 1}{60})}^{2}} \cdot 60 + \sqrt{1 + {(\frac{- 1}{120})}^{2}} \cdot 60 = 2 \sqrt{3601} + \sqrt{14401}$

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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 $y_{k} = 0, 1, 2, 3 . . . . . . . . n$ .
$f (x) = \cos x, [a, b] = [0, π], n = 3$

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find 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 yk=0,1,2,3........n.fx=cosx,a,b=0,π,n=3

Short Answer

Step by step solution

Step 1. Given information .

Step 2. Formula used .

Step 3. Find arc length .

One App. One Place for Learning.

Most popular questions from this chapter

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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 $y_{k} = 0, 1, 2, 3 . . . . . . . . n$ .
$f (x) = \cos x, [a, b] = [0, π], n = 3$