Use Simpson's Rule with n = 10 to estimate the arc length of the curve.
y = tan x, 0 < x < π/9
To estimate the arc length of the curve y = tan x, 0 < x < π/9 using Simpson's Rule with n = 10, we need to follow these steps:
Step 1: Determine the interval of integration
The interval of integration is given as 0 < x < π/9. This means we need to find the arc length on this interval.
Step 2: Calculate the step size
To use Simpson's Rule, we divide the interval into equally spaced subintervals. The step size is calculated as h = (b - a) / n, where n is the number of subintervals and a and b are the limits of integration. In this case, n = 10, a = 0, and b = π/9. Therefore, the step size is h = (π/9 - 0) / 10 = π/90.
Step 3: Evaluate the function at the endpoints and interior points
We need to evaluate the function y = tan x at the endpoints (0 and π/9) and the interior points within the subintervals. Since n = 10, there are 11 points in total.
Step 4: Calculate the weights
Simpson's Rule uses different weights for the endpoints, interior points, and midpoints of the subintervals. The weights for the endpoints are 1, while the weights for the interior points alternate between 4 and 2.
Step 5: Compute the approximate arc length
Using Simpson's Rule, the approximate arc length can be calculated as follows:
L ≈ h / 3 * (y₀ + 4y₁ + 2y₂ + 4y₃ + 2y₄ + 4y₅ + 2y₆ + 4y₇ + 2y₈ + 4y₉ + y₁₀),
where y₀ = y(a), y₁ = y(a + h), y₂ = y(a + 2h), y₃ = y(a + 3h), and so on.
In our case, the approximate arc length using n = 10 would be:
L ≈ (π/90) / 3 * (tan 0 + 4tan(π/90) + 2tan(2π/90) + 4tan(3π/90) + 2tan(4π/90) + 4tan(5π/90) + 2tan(6π/90) + 4tan(7π/90) + 2tan(8π/90) + 4tan(9π/90) + tan(π/9)).
Step 6: Calculate the value of the approximate arc length
Evaluate the expression using a calculator or software to obtain the numerical value of the approximate arc length.
It's important to note that Simpson's Rule is an approximation method and may not give an exact value for the arc length. However, using more subintervals (increasing n) can improve the accuracy of the approximation.