From 8ae196ae3d3692b6ef818b1009cc061662f0baab Mon Sep 17 00:00:00 2001 From: arc Date: Tue, 25 Mar 2025 09:11:37 -0600 Subject: [PATCH] vault backup: 2025-03-25 09:11:37 --- education/math/MATH1210 (calc 1)/Integrals.md | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/education/math/MATH1210 (calc 1)/Integrals.md b/education/math/MATH1210 (calc 1)/Integrals.md index bc24b75..8dbfbbf 100644 --- a/education/math/MATH1210 (calc 1)/Integrals.md +++ b/education/math/MATH1210 (calc 1)/Integrals.md @@ -35,7 +35,8 @@ $$ \int f(x) dx $$ | $\dfrac{d}{dx} \tan^{-1} x = \dfrac{1}{1+x^2}$ | $\int \dfrac{1}{\sqrt{1+x^2}}dx = \tan^{-1}x + C$ | | $\dfrac{d}{dx} k f(x) = k f'(x)$ | $\int k*f(x)dx = k\int f(x)dx$ | | $\dfrac{d}{dx} f(x) \pm g(x) = f'(x) \pm g'(x)$ | $\int (f(x) \pm g(x))dx = \int f(x) dx \pm \int g(x) dx$ | - +# Area Under a Curve +The area under the curve $y = f(x)$ can be approximated by the equation $\sum_{i = 1}^n f(\hat{x_i})\Delta x$ where $\hat{x_i}$ is any point on the interval $[x_{i - 1}, x_i]$, and the curve is divided into $n$ equal parts of width $\Delta x$ # Definite Integrals Let $f$ be a continuous function on the interval $[a, b]$. Divide $[a, b]$ into $n$ equal parts of width $\Delta x = \dfrac{b - a}{n}$ . Let $x_0, x_1, x_2, \cdots, x_3$ be the endpoints of the subdivision. @@ -43,4 +44,6 @@ The definite integral of $f(x)$ with respect to $x$ from $x = a$ to $x = b$ can $$ \int_{a}^b f(x) dx $$ And __can__ be defined as: -$$ \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i = 1}^n f(x_i)\Delta x$$ \ No newline at end of file +$$ \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i = 1}^n f(x_i)\Delta x$$ + +$f(x_i)$ is the *height* of each sub-interval, and $\Delta x$ is the change in the *x* interval, so $f(x_i) \Delta x$ is solving for the area of each sub-interval. \ No newline at end of file