Problems

Approximate the integrals in Problems 1-20 using (a) the Trapezoidal Rule and (b) Simpson's Rule. When possible, find error estimates. If a hand calculator is available, do the problems again with Ax = 0.1.

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19 20

21             Let f be continuous on the interval [a, b] and let Δx = (b - a)/n where n is a positive integer. Prove that the trapezoidal sum is equal to the Riemann sum plus ½(f(b) -f(a)) Δx, that is,

Show that if f(a) = f(b) then the trapezoidal sum and Riemann sum are equal.

22            Prove that for a linear function f(x) = kx + c, the trapezoidal sum is exactly equal to the integral.

23            Show that if f(x) is concave downward, f"(x) > 0, then the trapezoidal sum is less than the definite integral of f(x).

24            Show that for a quadratic function f(x) = ax² + bx + c, Simpson's approximation is equal to the definite integral.

25            Show that for a cubic function f(x) = ax³ + bx² + cx + d, Simpson's approximation is still equal to the definite integral.