224k views
2 votes
find the smallest value of n such that the error in approximating the integral is less than 0.000000001 when using simpson's rule.

1 Answer

2 votes

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.

User Majk
by
7.7k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.