How do the left-endpoint, right-endpoint, and midpoint rectangle rules differ?

Conditions

• The subintervals have the same width.
• The method approximates area under a curve by sums of rectangles.

Reasoning, step by step

1. Partition the interval into subintervals of equal width.
2. For the left-endpoint rule, take the function value at the left edge of each subinterval.
3. For the right-endpoint rule, take the function value at the right edge of each subinterval.
4. For the midpoint rule, take the function value at the center of each subinterval.
5. Calculate the area of each rectangle and sum them up.

Example

The lecture presents three basic rectangle rules for approximating an integral over $[a,b]$: the left-endpoint rule, the right-endpoint rule, and the midpoint rule. In all three, the interval is partitioned into subintervals of equal width, and the only difference is which sample point in each subinterval determines the rectangle height.

Common misconceptions

• Thinking the three rectangle rules differ because they use differently sized subintervals.
• Assuming the midpoint rule uses the average of the left and right endpoints.

Watch the explanation

Explore next

• Why does the lecturer recommend halving each interval when refining the step size?
• What is the trapezoidal rule and why do the endpoints get weight one half?