Left, Right, and Midpoint Riemann Sums

What a Riemann Sum Is

A definite integral measures accumulated change or signed area. But sometimes you are not given an antiderivative, or the AP Calculus problem specifically asks you to approximate the integral from a table or graph.

A Riemann sum approximates the area under a curve by dividing the interval into smaller pieces and replacing the curve on each piece with a rectangle.

The basic idea is simple: add up the areas of the rectangles. Every rectangle has width and height determined by a selected point in the interval.

The only question is: which point should determine the height? That's what separates left, right, and midpoint Riemann sums.

Step 1: Find the Width of Each Rectangle

If the interval is and it is divided into equal subintervals, then every rectangle has width

This value is one of the most important numbers in the problem. Find first, before calculating any heights.

Left Riemann Sums

For a left Riemann sum, use the left endpoint of every subinterval to determine the rectangle's height.

Suppose the interval is divided at

The left endpoints are

Therefore, the left Riemann sum is

Right Riemann Sums

For a right Riemann sum, use the right endpoint of every subinterval.

For the same partition, the right endpoints are

The right Riemann sum is therefore

Midpoint Riemann Sums

A midpoint Riemann sum uses the midpoint of each subinterval rather than either endpoint.

For a subinterval from to , the midpoint is

So if the subintervals are , the midpoint Riemann sum is

The Three Formulas Side by Side

For equally spaced subintervals of width , the three sums can be summarized as follows:

The easiest way to remember the distinction is:

• Left: use the left side of each interval.
• Right: use the right side of each interval.
• Midpoint: use the center of each interval.

Reading a Table Correctly

AP Calculus frequently gives you a table instead of an equation. In these problems, the most important skill is recognizing which rows or entries the Riemann sum actually needs.

Suppose a table contains values at :

If the problem asks for a left sum on , you use the values at , not every value in the table.

If it asks for a right sum, use .

If it asks for a midpoint sum, the table must provide the function values at the midpoints, or you must otherwise be able to determine them.

Overestimates and Underestimates

One of the most common AP Calculus questions asks whether a Riemann sum is an overestimate or underestimate of the actual integral.

The key is to think about the shape of the function.

If the Function Is Increasing

If is increasing, then the left endpoint gives a rectangle height that is generally too small, while the right endpoint gives a height that is generally too large.

So for an increasing function:

• Left sum → underestimate
• Right sum → overestimate

If the Function Is Decreasing

If is decreasing, the situation reverses:

• Left sum → overestimate
• Right sum → underestimate

What About Midpoint Sums?

Midpoint sums are usually more accurate because they sample the function near the center of each interval. However, you should not automatically call a midpoint sum an overestimate or underestimate just because it is a midpoint sum.

The direction of the error depends on the function's concavity.

For a concave-up function, the midpoint rectangles generally lie below the curve, producing an underestimate:

For a concave-down function, the midpoint rectangles generally lie above the curve, producing an overestimate:

A Fast AP Exam Strategy

When you see a Riemann-sum problem on the AP exam, use this sequence:

1. Identify the interval. Find the starting and ending values.
2. Count the subintervals. Determine how many rectangles are being used.
3. Find . Use .
4. Identify the sample points. Left, right, or midpoint?
5. Select the correct function values. Do not use unnecessary table entries.
6. Multiply the sum of the heights by .
7. Check the direction of the approximation if the problem asks for an overestimate or underestimate.

Common Mistakes

Practice Problems

The Big Picture

Riemann sums are the bridge between rectangle approximations and the definite integral. As the number of rectangles increases and their widths become smaller, the approximation becomes closer to the exact accumulated area.

For AP Calculus, the most important distinction is not memorizing three disconnected formulas. It is recognizing that every Riemann sum follows the same structure:

Choose the sample points, find the heights, add them, and multiply by the width. Once that pattern becomes automatic, table-based Riemann-sum questions become much faster.