Write down an integral expressing the length of the curve y = cos x, 2 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Problem Statement:**
Write down an integral expressing the length of the curve
\[ y = \cos x, \quad -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \]
Use technology to give an approximate value for this length. Do you have confidence in this value? Why or why not?
---
**Explanation:**
To calculate the length of a curve given by \( y = f(x) \) from \( x = a \) to \( x = b \), we use the formula for arc length:
\[
L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]
For the function \( y = \cos x \), the derivative is \( \frac{dy}{dx} = -\sin x \).
Therefore, the integral for the arc length becomes:
\[
L = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{1 + (-\sin x)^2} \, dx = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{1 + \sin^2 x} \, dx
\]
**Approximating the Length:**
Using technology such as a graphing calculator or computer software, compute this integral to obtain an approximate value for the curve's length.
**Reflection:**
Consider if the approximation is reliable. The confidence in the value depends on the numerical methods used by the technology, such as the precision of the calculations and the handling of any potential sources of error.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf89c864-71e5-4449-bde3-4774682a1969%2F41ac0563-430c-4b65-a47b-9a6415d9944a%2Fp9nim5g_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Write down an integral expressing the length of the curve
\[ y = \cos x, \quad -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \]
Use technology to give an approximate value for this length. Do you have confidence in this value? Why or why not?
---
**Explanation:**
To calculate the length of a curve given by \( y = f(x) \) from \( x = a \) to \( x = b \), we use the formula for arc length:
\[
L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]
For the function \( y = \cos x \), the derivative is \( \frac{dy}{dx} = -\sin x \).
Therefore, the integral for the arc length becomes:
\[
L = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{1 + (-\sin x)^2} \, dx = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{1 + \sin^2 x} \, dx
\]
**Approximating the Length:**
Using technology such as a graphing calculator or computer software, compute this integral to obtain an approximate value for the curve's length.
**Reflection:**
Consider if the approximation is reliable. The confidence in the value depends on the numerical methods used by the technology, such as the precision of the calculations and the handling of any potential sources of error.
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