When is the Mid-point rule is the worsted possible option for estimating area ( Riemann sum )?
The midpoint rule is one of the methods used to estimate the area under a curve, also known as a Riemann sum. It works by dividing the interval over which the area is being estimated into subintervals, and then approximating the area within each subinterval using the midpoint of that subinterval.
The midpoint rule is typically less accurate compared to other Riemann sum methods, such as the left endpoint rule or the right endpoint rule. The main reason for this is that it assumes the shape of the curve within each subinterval is best approximated by a straight line connecting the midpoint of the interval to the curve. This assumption may lead to larger errors when the curve is not relatively linear in each subinterval.
Thus, the midpoint rule tends to be the worst possible option for estimating the area when the curve being integrated is highly curved or has significant variations within each subinterval. In such cases, using other Riemann sum methods, such as the left or right endpoint rule, or more advanced numerical integration techniques like Simpson's rule or the trapezoidal rule, can provide more accurate estimations of the area.