Question

Estimate the minimum number of subintervals needed to approximate the value of ∫−33(5x2+7) dx with an error of magnitude less than 4×10−4 by a. the Trapezoidal Rule. b. Simpson's Rule. a. The minimum number of subintervals using the Trapezoidal Rule is (Round up to the nearest whole number.) b. The minimum number of subintervals using Simpson’s Rule is

Answer 1

To estimate the minimum number of subintervals needed for the Trapezoidal Rule, we use the formula h = (b - a) / n. For Simpson's Rule, we use the formula h = (b - a) / (2n).

Step-by-step explanation:

To estimate the minimum number of subintervals needed for the Trapezoidal Rule, we can use the formula h = (b - a) / n, where h is the width of each subinterval, b is the upper limit of integration, a is the lower limit of integration, and n is the number of subintervals. Rearranging the formula, we get n = (b - a) / h. Here, b = 3, a = -3, and we want the error to be less than 4x10^-4. Let's assume n is an integer, so we'll round it up to the nearest whole number. Substituting the values, we get n = (3 - (-3)) / h = 6 / h. To find the minimum number of subintervals, we need to find the maximum value of h that satisfies the given error condition.

For Simpson's Rule, we use the formula h = (b - a) / (2n), where h is the width of each subinterval and n is the number of subintervals. Rearranging the formula, we get n = (b - a) / (2h). Here, b = 3, a = -3, and we want the error to be less than 4x10^-4. Let's assume n is an integer, so we'll round it up to the nearest whole number. Substituting the values, we get n = (3 - (-3)) / (2h) = 6 / (2h). To find the minimum number of subintervals, we need to find the maximum value of h that satisfies the given error condition.