The table gives the values of a function obtained from an ex

The table gives the values of a function obtained from an experiment. Use them to estimate f(x)dx using three equal subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. If the function is known to be a decreasing function, can you say whether your estimates are less than or greater than the exact value of the integral?

Solution

(1) The right hand approximation is 2(8.3 + 2.3 + (-10.5)) = 0.2. (2) The left hand approximation is 2(9.3 + 8.3 + 2.3) = 39.8. (3) The midpoint approximation is 2(9 + 6.5 + (-7.6)) = 15.8. The right hand Riemann sum will always give an underestimate to the integral. The left hand Riemann sum will overestimate the integral for a decreasing function. For the midpoint rule you cannot tell.