AP® Calculus AB · Unit 6 · Topic 6.2
Left, Right, and Midpoint Sums
Left, right, and midpoint sums all approximate the area under a curve by adding up rectangle areas, but they differ in where each rectangle's height is measured, which decides whether the estimate runs high or low.
A Riemann sum cuts the interval into equal pieces and builds a rectangle on each one. The only choice is where you sample the height: at the left edge, the right edge, or the midpoint of the subinterval. That choice decides whether the rectangle's flat top sits above or below the curve, which is the whole source of the error. On a rising function, the left edge is the lowest point of each piece, so left rectangles sit under the curve and the left sum undershoots. The right edge is the highest point, so the right sum overshoots. Flip the function to decreasing and the pattern flips too: left becomes the high point, right becomes the low point.
The midpoint sum behaves differently because it is not just splitting the difference between left and right. By sampling the center, it lets the curve's bend work for and against the rectangle at the same time: on one half of the subinterval the curve dips below the rectangle's top, and on the other half it rises above, and those errors partly cancel. That is why midpoint is usually closer to the true area than the simple average of the left and right sums, which is really the trapezoid estimate and only accounts for slope, not curvature. The direction of concavity controls how well midpoint cancels its error.
These sums are the setup for the definite integral: as the number of rectangles grows without bound, all three sums converge to the same exact area, which the Fundamental Theorem of Calculus lets you compute directly.
Key terms
- Riemann sum
- An approximation of the area under a curve using a sum of rectangle areas over subintervals.
- Left sum
- A Riemann sum where each rectangle's height comes from the function value at the left edge of its subinterval.
- Right sum
- A Riemann sum where each rectangle's height comes from the function value at the right edge of its subinterval.
- Midpoint sum
- A Riemann sum where each rectangle's height comes from the function value at the center of its subinterval.
- Trapezoid rule
- An estimate equal to the average of the left and right sums, which models each piece as a trapezoid instead of a rectangle.
Watch it explained
- How it worksHow do the left, right, and midpoint choices shift an estimate up or down?
- Worked exampleFind the left and right Riemann sums for f(x) = 1/x on 1 to 3 with four equal pieces.
- Common mistakeWhy isn't the midpoint sum the average of the left and right sums?
- Going furtherWhat happens to the over- and under-estimate pattern when the function is falling?
Check yourself
End of topic.