From adcf1012eb594403cb1e5b65f7e4e6a6c14a86fd Mon Sep 17 00:00:00 2001 From: arc Date: Thu, 17 Apr 2025 09:25:41 -0600 Subject: [PATCH] vault backup: 2025-04-17 09:25:41 --- education/math/MATH1210 (calc 1)/Integrals.md | 11 ++++++++++- 1 file changed, 10 insertions(+), 1 deletion(-) diff --git a/education/math/MATH1210 (calc 1)/Integrals.md b/education/math/MATH1210 (calc 1)/Integrals.md index 12ab62a..89082f7 100644 --- a/education/math/MATH1210 (calc 1)/Integrals.md +++ b/education/math/MATH1210 (calc 1)/Integrals.md @@ -107,6 +107,9 @@ This formula can also be stated as $\int_a^b f(x)dx = f(c)(b-a)$ This theorem tells us that a continuous function on the closed interval will obtain its average for at least one point in the interval. # U-Substitution +When you see $dx$ or $du$ in a function, it can be thought of as roughly analogous to $\Delta x$, where the change in $x$ is infinitesimally small. + +Thinking back to derivatives, when solving for $\dfrac{dy}{dx}$, you're solving for the rate of change of $y$ (across an infinitely small distance) over the rate of change of $x$ (across an infinitely small i$nstance). Given that the *slope* of a line is described as $\dfrac{\text{rise}}{\text{run}}$, ## Formulas - $\int k {du} = ku + C$ - $\int u^n du = \frac{1}{n+1}u^{n+1} + C$ @@ -204,4 +207,10 @@ The height of the rectangle, or the distance between the curves at a given point The Riemann Sum definition of the area between two curves is as follows: -$$ \lim_{n \to \infty} \sum_{i = 0}^n (f(x_i)-g(x_i)\cdot \Delta x)$$ \ No newline at end of file +$$ \lim_{n \to \infty} \sum_{i = 1}^n (f(x_i)-g(x_i)\cdot \Delta x)$$ +- $i$ is the sub-interval +- $x_i$ is the $x$ coordinate at a given sub-interval +- $\Delta x$ is the width of each sub-interval. + +This sum can also be described as: +$$ = \int_a^b $$ \ No newline at end of file