use the simpson's rule with n=10 to estimate arc length of y=x^(-1/3), for 1<=x<6
http://www.intmath.com/integration/6-simpsons-rule.php
To use Simpson's Rule to estimate the arc length of a function, we first need to obtain a set of coordinates by evaluating the function at equally spaced values of x within the given interval. Then, we can use Simpson's Rule formula to estimate the arc length.
1. Divide the interval [1, 6] into n subintervals. Since n = 10, each subinterval will have a width of (6 - 1) / 10 = 0.5. Therefore, we get the following equally spaced x-values:
x0 = 1.0
x1 = 1.5
x2 = 2.0
x3 = 2.5
x4 = 3.0
x5 = 3.5
x6 = 4.0
x7 = 4.5
x8 = 5.0
x9 = 5.5
x10 = 6.0
2. Evaluate the function y = x^(-1/3) at these x-values to obtain a set of y-values:
y0 = (x0)^(-1/3)
y1 = (x1)^(-1/3)
y2 = (x2)^(-1/3)
...
y9 = (x9)^(-1/3)
y10 = (x10)^(-1/3)
3. Apply Simpson's Rule formula to estimate the arc length:
Arc length ≈ (h/3) * [y0 + 4y1 + 2y2 + 4y3 + 2y4 + 4y5 + 2y6 + 4y7 + 2y8 + 4y9 + y10]
where h is the width of each subinterval, which is 0.5 in this case.
Substituting the values we obtained:
Arc length ≈ (0.5/3) * [y0 + 4y1 + 2y2 + 4y3 + 2y4 + 4y5 + 2y6 + 4y7 + 2y8 + 4y9 + y10]
4. Calculate the values of y0 to y10 using the corresponding x-values obtained earlier.
5. Substitute the calculated values of y0 to y10 into the formula and perform the arithmetic calculations to find the estimated arc length of the function.