In general, for approximate integration: - left end-point approximation (left Riemann's Sum) gives an UNDERESTIMATE - right end-point approximation (right Riemann's Sum) gives an OVERESTIMATE - mid-point rule gives an UNDERESTIMATE. Is this correct?
If the function is increasing and is increasing at an increasing rate (such as the second derivative is positive), then yes. But these statements are not universally true of Riemann integrable functions. For example, the constant function f(x) = 1. In the case, the Riemann sum approximations are all exact. Another example, y =1-x, integration from x=0 to 1. Then the left end point approximations are going to be high, the right end ones low, and the mid-points will be exact.
Ok I understand. Thank you.
@JamesJ Unfortunately, this function was increasing at an increasing rate and I said that the midpoint rule would give an underestimate, but this was wrong. Why?
that function is increasing at a decreasing rate (concave down)
Makes sense. Thanks