Example 1 Estimate by the six Rectangle Rules using the regular

1. Example 1 Estimate by the six Rectangle Rules using the regular partition P of the interval [0,1] into 4 subintervals. Solution This definite integral gives the area of a quarter circle of radius 1 and therefore has value /4. Hence estimating this integral means estimating the value of /4. Note that P = {0, ¼, ½, ¾, 1}with each subinterval of width ¼. The four subintervals are: [0, ¼ ], [¼, ½], [½,¾] and [¾, 1] while the function is Then The Lkare the heights of the 4 rectangles used to approximate this definite integral. In the Left Endpoint Rule the Lk are the values of f on the left endpoints of the four subintervals: f(0), f(¼), f(½), f(¾). In the Right Endpoint Rule the Lk are the values of f on the right endpoints of the four subintervals: f(¼), f(½), f(¾), f(1).

2. The 4 subintervals are: [0, ¼ ], [¼, ½], [½,¾], [¾, 1]. In the Midpoint Rule the Lk are the values of f on the midpoints of the four subintervals: f(1/8), f(3/8), f(5/8), f(7/8). In the Trapezoid Rule the Lk are the averages of the values of f on the endpoints of each of the four subintervals: ½[f(0)+f(¼)], ½[f(¼)+f(½)], ½[f(½)+f(¾)], ½[f(¾)+f(1)]. Therefore, the Lower Riemann sum coincides with the estimate of the Right Endpoint Rule and the Upper Riemann sum coincides with estimate of the Left Endpoint Rule. Since the function f is decreasing on [0,1], it has its maximum value at the left endpoint of each subinterval and its minimum value at the right endpoint of each subinterval. The values of the Lkare summarized in the table on the next slide

3. The Left Endpoint Rule and Upper Riemann Sum give the same estimate: The Right Endpoint Rule and Lower Riemann Sum give the same estimate: The Midpoint Rule gives the estimate: The Trapezoid Rule gives the estimate: