The statement "If f is a linear function on the interval [a, b], then a midpoint Riemann sum gives the exact value of Integral from a to b f (x )dx for any n" is false.
The statement is false. The midpoint Riemann sum gives the exact value of the definite integral of a function f over an interval [a, b] only if f is linear over that interval.
For other functions, the midpoint Riemann sum is an approximation that becomes more accurate as the number of subintervals increases.
Consider the function f(x) = x^2 over the interval [0, 1].
This function is not linear, and the midpoint Riemann sum with n = 100 subintervals gives the value 0.3333, while the actual value of the definite integral is 0.33333333....
Therefore, the statement "If f is a linear function on the interval [a, b], then a midpoint Riemann sum gives the exact value of Integral from a to b f (x )dx for any n" is false.
Question
If f is a linear function on the interval [a,b], then a midpoint Riemann sum gives the exact value of Integral from a to b f (x )dx for any n. Is the statement true or false? Explain.