Question

Suppose f(x) is a decreasing, positive function, and we use different methods to approximate the integral of f over the interval [a, b]. Let U_f(p) denote the upper sum and L_f(p) denote the lower sum with respect to a partition p of 90 subintervals with equal length. Let m_90 represent the Riemann sum with the midpoint approach with 90 subintervals of equal length. Which of the following statements is certainly true?

a) U_f(p) > m_90 > L_f(p)
b) L_f(p) > m_90 > U_f(p)
c) U_f(p) = L_f(p) = m_90
d) The relationship between U_f(p), L_f(p), and m_90 cannot be determined.

Answer 1

The statement that is certainly true is a) U_f(p) > m_90 > L_f(p).

Step-by-step explanation:

The statement that is certainly true is a) U_f(p) > m_90 > L_f(p).

Here's why:

1. Since the function is decreasing and positive, the upper sum U_f(p) will be the sum of the maximum values of f(x) in each subinterval.
2. Similarly, the lower sum L_f(p) will be the sum of the minimum values of f(x) in each subinterval.
3. With the midpoint approach, the Riemann sum m_90 will be the sum of the values of f(x) at the midpoints of each subinterval. Since the function is decreasing, the values at the midpoints will be between the maximum and minimum values in each subinterval.

Therefore, U_f(p) > m_90 > L_f(p).