Rectangular Formula for Definite Integrals

Theorem

Let $f$ be a real function which is integrable on the closed interval $\closedint a b$.

Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\closedint a b$:

$\forall r \in \set {1, 2, \ldots, n}: x_r - x_{r - 1} = \dfrac {b - a} n$

Then the definite integral of $f$ with respect to $x$ from $a$ to $b$ can be approximated as:

$\ds \int_a^b \map f x \rd x \approx \sum_{r \mathop = 0}^{n - 1} h \map f {x_r}$

where $h = \dfrac {b - a} n$.

Proof

This theorem requires a proof. In particular: This will probably boil down to a graphical approach based on the structure of a Darboux integral. We need to explain rigorously what an "approximation" means, and we may also want to quantify the error. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Trapezium Rule for Definite Integrals
• Simpson's Rule

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Approximate Formulas for Definite Integrals: $15.15$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 18$: Definite Integrals: Approximate Formulas for Definite Integrals: $18.15$