Estimate the length of the curve f(x)= cos x on 0 ≤ x ≤ pi/2. Using n=4

To estimate the length of the curve \( f(x) = \cos(x) \) on the interval \( 0 \leq x \leq \frac{\pi}{2} \) using \( n = 4 \) subintervals, we can apply the **arc length formula** for a function \( y = f(x) \):

\[
L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]

1. Given \( f(x) = \cos(x) \), the derivative is:
   \[
   \frac{dy}{dx} = -\sin(x)
   \]

2. Substituting into the arc length formula:
   \[
   L = \int_0^{\frac{\pi}{2}} \sqrt{1 + (-\sin(x))^2} \, dx = \int_0^{\frac{\pi}{2}} \sqrt{1 + \sin^2(x)} \, dx
   \]

3. We approximate this integral using the **trapezoidal rule** with \( n = 4 \) subintervals over the interval \( [0, \frac{\pi}{2}] \). The step size is:
   \[
   \Delta x = \frac{\frac{\pi}{2} - 0}{4} = \frac{\pi}{8}
   \]

4. The x-values at which we evaluate the function are:
   \[
   x_0 = 0, \quad x_1 = \frac{\pi}{8}, \quad x_2 = \frac{\pi}{4}, \quad x_3 = \frac{3\pi}{8}, \quad x_4 = \frac{\pi}{2}
   \]

5. The corresponding function values at these points are:
   \[
   f(x_0) = \sqrt{1 + \sin^2(0)} = 1, \quad f(x_1) = \sqrt{1 + \sin^2\left( \frac{\pi}{8} \right)}
   \]
   and so on for each \( x \).

Let me now calculate the length using the trapezoidal rule approximation.The estimated length of the curve \( f(x) = \cos(x) \) on the interval \( 0 \leq x \leq \frac{\pi}{2} \) using \( n = 4 \) subintervals is approximately \( 1.91 \) units.

Would you like further details on the method or any specific steps explained?

Here are 5 related questions for deeper exploration:
1. How does increasing the number of subintervals \( n \) affect the accuracy of the arc length approximation?
2. What is the exact arc length for this curve using analytical methods?
3. How does the Trapezoidal rule compare to Simpson's rule for arc length approximation?
4. How would you calculate the length of the curve \( f(x) = \sin(x) \) over the same interval?
5. Can the arc length formula be applied to parametric curves? How would that differ?

**Tip:** When using numerical methods like the trapezoidal rule, increasing the number of subintervals improves the accuracy of the approximation, but may require more computation time.

Estimate the length of the curve f(x) = cos(x) on the interval 0 ≤ x ≤ π/2 using the trapezoidal rule with 4 subintervals. This problem explores key calculus concepts such as arc length and numerical integration techniques. Suitable for undergraduate calculus students.

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