Question

Suppose T is the approximation of this area using 15 trapezoids of equal width, R is the Riemann sum using right-hand endpoints and 15 subdivisions, and A is the true area of this region. Which of the following is correct?

A. T will always be greater than R.

B. R will always be greater than T.

C. T and R will be equal.

D. The relationship between T and R cannot be determined from the information given.

Answer 1

The relationship between trapezoidal approximation T and Riemann sum R cannot be determined without additional details about the curve shape; hence, neither will always be greater than the other. The correct answer is option D. The relationship between T and R cannot be determined from the information given.

Step-by-step explanation:

The question you've asked pertains to the approximation of the area under a curve compared to the actual area A, using 15 trapezoids (T) and a Riemann sum with right endpoints (R). The relationship between T, R, and A depends on the specific function and the curve's shape. Trapezoidal approximation T overestimates the area for concave down functions and underestimates for concave up functions. Conversely, a Riemann sum with right endpoints will tend to overestimate areas where the curve is increasing and underestimate where it's decreasing.

Therefore, without additional information about the shape of the curve, we cannot definitively say that T will always be greater than R or vice versa. So the correct answer is D: The relationship between T and R cannot be determined from the information given.