Find the Error resulted from approximation by Simpson's Rule:

integral (from 0 to 1) sqrt( 1+x^3) dx
... compute the result for n=8

LOL you can do this one, i just did the trapezoid

here

1_ 4_ _2_ _4__ 2_ _4_ _2__ 4_ 1
* values of (1+x^3)^.5 at
0 1/8 1/4 3/8 1/2 5/8 3/4 7/8 1
You will find those y values at the trapezoid problem that I already did

and use (1/8)/3 = 1/24 times sum

I am a Naval Architect. I wish I had seen this before the trapezoid one. We use Simpsons for everything because ship curves tend to be sort of parabolic.

To find the error resulting from approximation by Simpson's Rule, we need to calculate the error term for the given integral.

Simpson's Rule states that the error term E for approximating an integral using Simpson's Rule with n subintervals is given by the formula:

E = (-1/90) * [(b-a) / (2n)^4] * f''''(c)

where:
- a and b are the limits of integration
- n is the number of subintervals
- f''''(c) represents the fourth derivative of the function evaluated at some point c in the interval [a, b]

In this case, we are approximating the integral of sqrt(1+x^3) from 0 to 1 using Simpson's Rule with n = 8.

First, we need to compute f''''(x). Let's start by finding the derivatives of the function:

f(x) = sqrt(1 + x^3)

f'(x) = (3/2) * x^(1/2)

f''(x) = (3/4) * x^(-1/2)

f'''(x) = (-3/8) * x^(-3/2)

f''''(x) = (9/16) * x^(-5/2)

To evaluate f''''(c), we need to find the value of c.

Now, let's compute the integral using Simpson's Rule with n = 8:

Using Simpson's Rule, the integral can be approximated as:

Integral ≈ [x0*f(x0) + 4(x1*f(x1) + x2*f(x2)) + ... + 4(x6*f(x6) + x7*f(x7)) + x8*f(x8)] * (h/3)

where:
- x0 = 0, x1 = h, x2 = 2h, ..., xn = nh are the equally spaced subinterval values
- h = (b-a)/n is the width of each subinterval

Since we have n = 8, h = (1-0)/8 = 1/8 = 0.125

x0 = 0
x1 = 0.125
x2 = 0.25
x3 = 0.375
x4 = 0.5
x5 = 0.625
x6 = 0.75
x7 = 0.875
x8 = 1

Now, substitute the values into the formula:

Integral ≈ [0*f(0) + 4(0.125*f(0.125) + 0.25*f(0.25)) + ... + 4(0.75*f(0.75) + 0.875*f(0.875)) + 1*f(1)] * (0.125/3)

Evaluate f(x) at each of these subinterval values and calculate the sum:

Integral ≈ [0 + 4(0.125*f(0.125) + 0.25*f(0.25)) + ... + 4(0.75*f(0.75) + 0.875*f(0.875)) + f(1)] * (0.125/3)

Finally, calculate the error term using the formula mentioned above:

E = (-1/90) * [(b-a) / (2n)^4] * f''''(c)

Substitute the respective values:

E = (-1/90) * [(1-0) / (2*8)^4] * f''''(c)

Solve this expression to find the error resulted from the approximation using Simpson's Rule with n = 8.