use the simpsons rule with n=10, arc length y=^3√x, for 1<=x<6
fix your equation first
I think she means y = ∛x
y = ∛x for 1 is less then or equal to x and x is less then 6
I assume that you know the formula ....
in case you don't, here is the method
http://www.mathwords.com/s/simpsons_rule.htm
∆x = (6-1)/3 = 5/3 , so ∆x/3 = 5/9
interval = (6-1)/10 = 1/2
then we have
(5/9)[ f(1) + 4f(1.5) + 2f(2) + 4f(2.5) + 2f(3) + 4f(3.5) + 2f(4) + 4f(4.5) + 2f(5) + 4f(5.5) + f(6) ]
= (5/9)[ 1 + 4.5789 + 2.5198 + ... + 7.0607 + 1.81712 ]
I will let you do the rest of the button pushing.
∆x = (6-1)/3 = 5/3 , so ∆x/3 = 5/9
should be
∆x = (6-1)/10 = 5/3 , so ∆x/3 = 5/30 = 1/6
and the 5/3 in front of the big brackets should be 5/30 or 1/6
aprox 10.2688 thank you so much
To use Simpson's Rule, we need to approximate the definite integral of a function over a given interval. In this case, we want to find the arc length of the function y = ∛x over the interval 1 ≤ x < 6.
Simpson's Rule is based on approximating the function with quadratic polynomials on subintervals of equal width. The formula for Simpson's Rule can be written as:
S ≈ h/3 * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + 4f(xn-1) + f(xn)]
Where:
- h is the width of each subinterval, given by h = (b - a) / n, where a and b are the limits of integration, and n is the number of subintervals.
- x0, x1, x2, ..., xn are the equally spaced points within the interval.
In our case, a = 1, b = 6, and n = 10. So, we have:
- h = (6 - 1) / 10 = 0.5
- x0 = 1
- x1 = 1 + 0.5 = 1.5
- x2 = 1.5 + 0.5 = 2
- x3 = 2 + 0.5 = 2.5
- x4 = 2.5 + 0.5 = 3
- x5 = 3 + 0.5 = 3.5
- x6 = 3.5 + 0.5 = 4
- x7 = 4 + 0.5 = 4.5
- x8 = 4.5 + 0.5 = 5
- x9 = 5 + 0.5 = 5.5
- xn = b = 6
Now, we need to evaluate the function ∛x at these points:
- f(x0) = ∛1
- f(x1) = ∛1.5
- f(x2) = ∛2
- f(x3) = ∛2.5
- f(x4) = ∛3
- f(x5) = ∛3.5
- f(x6) = ∛4
- f(x7) = ∛4.5
- f(x8) = ∛5
- f(x9) = ∛5.5
- f(xn) = ∛6
Now we can plug these values into Simpson's Rule formula:
S ≈ (0.5/3) * [∛1 + 4∛1.5 + 2∛2 + 4∛2.5 + 2∛3 + 4∛3.5 + 2∛4 + 4∛4.5 + 2∛5 + ∛5.5 + ∛6]
Calculate this sum using the given values to find the approximation for the arc length of ∛x over the interval 1 ≤ x < 6.