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Midpoint approximation method

This original textbook reading covers part of the AP topic. Follow the AP course outline for complete topic instruction and practice.

Midpoint approximation method

A method of estimating areas under curves. It is structured so that the midpoint of the top of each rectangle lies on the curve whose area is being approximated.

Example: Find the area of the curve $y = x^{2}$ from the interval 0.5 to 2.5 using the midpoint method, and n = 2 rectangles.

$Δ x = \frac{2.5 - 0.5}{n = 2} = 1$

$\begin{aligned} A & = f (0.5 + \frac{1}{2} Δ x) \cdot Δ x \\ + f (0.5 + Δ x + \frac{1}{2} Δ x) \cdot Δ x \end{aligned}$

$A = 1^{2} \cdot 1 + 2^{2} \cdot 1$

$A = 5$

Source figure

Index

Chapter 7 - Midpoint Method

Chapter 7 - Trapezoidal Method

Chapter 7 - FFR - Anti-derivatives

Glossary and deeper reading · 1 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Midpoint rule
The midpoint rule approximates a definite integral by sampling at the center of each subinterval. Each sampled height is multiplied by its own subinterval width. For example, one midpoint rectangle for x² on [0,2] has height 1 and width 2, giving 2. For a concave-up function the midpoint sum underestimates the integral, while for a concave-down function it overestimates it.
Read more: 6.2 · Midpoint rule →

Midpoint rule

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