Question

Which of the following statements are true? A) If the number of terms in a Riemann sum increases then the quantity Δ decreases. B) When the overestimate equals the underestimate you have found the exact area under the curve.

Answer 1

True, increasing the number of terms in a Riemann sum leads to a decrease in Δ, leading to a more accurate approximation. However, it's false that an equal overestimate and underestimate yield the exact area; this is only a specific case and not a general rule.

Step-by-step explanation:

The question pertains to Riemann sums and the estimation of areas under curves in calculus. The correct answers to the statements provided are:

A) True: If the number of terms in a Riemann sum increases, then the quantity Δ (delta), which represents the width of each rectangle, decreases. This results in a more precise approximation of the area under the curve because the rectangles can fit the curve more closely.

B) False: When the overestimate equals the underestimate, it does not necessarily mean you have found the exact area under the curve. This is a special case that may happen with certain symmetrical functions but is not a general rule for all integrals. The exact area under the curve is typically found by taking the limit of the Riemann sums as Δ approaches zero, not by merely comparing overestimates and underestimates.