A method of [[Quadrature]] such that for each step we aproximate the [[Integration|Integral]] using rectangles that match the [[Function]] $f$ at its *left* endpoint. ![[Pasted image 20260223154002.png|invert_B]] $\begin{align} A = hf(a)+h(a+h)+hf(a+2h) + \cdots + hf(b-h) \end{align}$ Where we draw rectangles of width $h$ from $a$ to $b$. $\huge \begin{align} \int_{a}^{b} f(x)\d x \approx h \sum_{i=0}^{ \pa{\frac{b-a}{h} -1}} f(a+ih) \end{align}$ For smaller values of $h$, this will will be more accurate. ### [[Numerical Analysis Error Bounds|Error Bound]] This method tends to be less accurate when the [[Derivative]] of $f$ is high. For a trivial example, $f(x)=kx$ $\large \begin{align} \int_{0}^{h}f(x)\d x &= \int_{0}^{h}kx\d x = \frac{kh^{2}}{2} \end{align}$ $\huge \begin{align} \epsilon = \frac{h^{2}}{f'} \end{align}$ ![[Left-endpoint Method .excalidraw.svg]] %%[[Left-endpoint Method .excalidraw.md|🖋 Edit in Excalidraw]]%% To extend to non-linear functions, The error for one step is $\le \frac{h^{2}}{2}f'_{ext}$, where $f'_{ext}$ is the largest absolute value of $f'$ in the interval. $\huge \begin{align} \epsilon &\leq \frac{h^{2}}{2}f'_{\op{ext}} \\ f'_{\op{ext}} &= \max_{x\in[a,b]}|f'(x)| \end{align}$ $\huge \frac{h(b-a)}{2}f'_{\op{ext}} $ #### Formal Proof Let $g(x)$ be such that $g'(x)=f(x)$ and $g(a)=0$. [[Universal Quantifier|For any]] $x\in [a,b]$, the [[Truncated Taylor Series]] is: $\huge \begin{align} g(x)&= g(a)+ (x-a)g'(x) + \frac{1}{2}(x-a)^{2}g''(\xi) \\ \xi &\in [a,b] \end{align}$