6.2 Approximating Areas with Riemann Sums

What else do you need to know about Approximating Areas with Riemann Sums?

You must handle both uniform and nonuniform partitions. A key requirement is analyzing the behavior of the function (such as increasing/decreasing or concavity) to determine and justify whether a given approximation method yields an underestimate or an overestimate of the true definite integral. The use of technology to perform these numerical approximations is also expected where appropriate.

Can you do these?

• I can approximate definite integrals for functions represented graphically, numerically, analytically, and verbally.
• I can approximate definite integrals using left, right, and midpoint Riemann sums, as well as trapezoidal sums, with uniform or nonuniform partitions.

How is Approximating Areas with Riemann Sums tested on the exam?

On the AP Exam, you may be asked to approximate definite integrals using data from tables (often with nonuniform partitions) or graphs. When using a calculator to evaluate or approximate a definite integral in free-response questions, you must clearly write the setup (the definite integral being evaluated, including endpoints and differential) that leads to the calculator's result. A common pitfall is failing to justify whether an approximation is an over- or underestimate; you must explicitly connect the approximation method to the relevant behavior of the function (e.g., 'because f(x) is strictly increasing, the right Riemann sum is an overestimate'). You should also be careful to show the sum of the products of the widths and heights when calculating Riemann sums from a table to communicate their method clearly.