Final answer:
To find the smallest value of n such that the error in approximating the integral is less than 0.000000001 using Simpson's rule, you need to calculate the error bound and solve for n.
Step-by-step explanation:
Simpson's rule is a numerical integration method used to approximate definite integrals. The formula for Simpson's rule is:
I = (h/3) * (y0 + 4y1 + 2y2 + ... + 2yn-2 + 4yn-1 + yn)
where h is the width of each interval and y0, y1, ..., yn are the function values at evenly spaced points.
To find the minimum value of n such that the error is less than 0.000000001, you need to calculate the error bound using the formula:
Error Bound = (1/180) * ((b-a)/n)^4 * M
where a and b are the limits of integration and M is the maximum value of the fourth derivative of the function over the interval [a, b].
By solving the equation Error Bound < 0.000000001 for n, you can find the smallest value of n.