Using the Trapezium rule with four ordinates (three strips), estimate to 4 significant figures the integral from 1 to 4 of (x^3+12)/4sqrt(x). Calculate the exact value of this integral, comparing it with your estimate. How could the estimate be improved?

Taking the value at the ordinates f(1) = 13/4, f(2) = 20/4sqrt(2), f(3) = 39/4sqrt(3), f(4) = 9.5 Then the trapezium rule states the integral is approximately 1/2 * [f(1) + 2f(2) + 2f(3) + f(4)], which (using a calculator) is 15.54 to 4 significant figures

(x^3+12)/4sqrt(x) = (1/4)x^(5/2) + 3x^(-1/2) Therefore the antiderivative F(x) = (1/14)x^(7/2) + 6x^(1/2) (as 1/14 = ((1/4)/(5/2 + 1)), 3 = (6/(-1/2 + 1))) And the exact value of the integral is F(4)-F(1) = 211/14 Using a calculator, the difference (211/14)-15.54 = -0.47 (to 2.s.f) This is rather inaccurate, estimate can be improved by using more ordinates.