How large should n be to guarantee that Simpson's Rule approximation to this integral is accurate to within 0.00001?

Question

How large should n be to guarantee that Simpson's Rule approximation to ∫(0 to 1)e^{x^2}dx is accurate to within 0.00001?

Answer 1

Let K = maximum value of l f''''(x) l on the interval [0,1].

f(x) = e^{x^2}

f'(x) = 2xe^{x^2}

f"(x) = 2e^{x^2} + 4x²e^{x^2} = e^{x^2}(2+4x²)

f'''(x) = 8xe^{x^2} + e^{x^2}(4x + 8x³) = e^{x^2}(12x + 8x³)

f""(x) = (12 + 24x²)e^{x^2} + e^{x^2}(24x² + 16x⁴) = e^{x^2}(12 + 48x² + 16x⁴)

K = l f""(1) l = 76e

So, we need to find n so that 76e(1-0)⁵ / (180n⁴) < 0.00001

1.147719 < 0.00001n⁴

114771.9< n⁴

So, n > 18.4

Recall that, with Simpson's Rule, n must be an even positive integer.

So, n = 20 works.