Question 1

Approximate the integral of f(x) = e^-x on [0, 10] using Simpson's Rule and Simpson's 3/8 Rule.

Answer: 1.711661979876841 and 1.388606601719423.

Question 2

Approximate the integral of f(x) = x² on the interval [0, 2] using Simpson's Rule and Simpson's 3/8 Rule.

Answer: 8/3 (or 2.666666666666667) in both cases.

Approximate the integral of f(x) = x⁴ on the interval [0, 2] using Simpson's Rule and Simpson's 3/8 Rule.

Answer: 20/3 = 6.666666666666667 and 176/27 = 6.518518518518516.

What is the estimated error for the approximation in Question 3 what is the actual error? Is this exact, and why?

Answer: Because the 4th derivative is a constant, the errors may be found exactly: -4/15 and -16/135.