Suppose we want to find the area under the curve (pictured below) on the interval using a Riemann sum with rectangles.

Which of the following provides the most accurate approximation of the area under the curve on the given interval?

Question

Suppose we want to find the area under the curve  (pictured below) on the interval  using a Riemann sum with  rectangles.

Which of the following provides the most accurate approximation of the area under the curve on the given interval?

BlackTom AI · Accepted Answer

The question asks about approximating the area under the given curve on a specified interval using a Riemann sum with n rectangles.
Option: midpoints
- Using rectangle heights evaluated at the midpoint of each subinterval generally yields a more accurate approximation than heights taken at the left or right endpoints when the function is not linear, especially for convex or concave shapes.
- In many cases, the midpoint rule has an error term that decreases faster with increasing n (often proportional to 1/n^2 for well-behaved functions) compared to the left- or right-Riemann sums, which typically have error proportional to 1/n for monotone or convex/concave sections.
- The pictured curve appears to be increasing and curved upward (convex), so sampling at midpoints tends to balance over- and under-estimation within each subinterval, reducing the overall error more effectively than using endpoints.
- Other potential options (not shown here) such as left-endpoint or right-endpoint sums would systematically under- or overestimate portions of the area for a monotone increasing, convex function, leading to larger error for the same number of rectangles.
- Therefore, among the typical rectangle-based Riemann sums, the midpoint approach is the conventionally more accurate choice for this kind of curve.
Note: If the interval, n, or function behavior changes, the relative accuracy could shift, but with a smooth increasing convex f(x) and equal-width subintervals, midpoints usually outperform endpoints for the same n.

MATH_1225_64262_202506 5.1 Areas and Distances

Suppose we want to find the area under the curve (pictured below) on the interval using a Riemann sum with rectangles. Which of the following provides the most accurate approximation of the area under the curve on the given interval?

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