Errors in the Trapezoidal Rule and Simpson's Rule

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If f has a continuous second derivative on [a, b], then the error E in approximating ∫[b to a] f(x) dx by the Trapezoidal Rule is |E| ≤ (b-a)^3/12n^2[max|f”(x)|], a<x<b. Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating ∫[b to a] f(x) dx by Simpson's Rule is |E| ≤ (b − a)^5/180n^4[max|f(4)(x)|], a ≤ x ≤ b. Use these to find the minimum integer n such that the error in the approximation of the definite integral is less than or equal to 0.

9.5: Trapezoidal and Midpoint Approximations