From d012cd70aabcd2a0a821b3a1e0e6babaaa0d6005 Mon Sep 17 00:00:00 2001 From: arc Date: Thu, 17 Apr 2025 09:10:41 -0600 Subject: [PATCH] vault backup: 2025-04-17 09:10:40 --- education/math/MATH1210 (calc 1)/Integrals.md | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/education/math/MATH1210 (calc 1)/Integrals.md b/education/math/MATH1210 (calc 1)/Integrals.md index 3c153f2..c29bace 100644 --- a/education/math/MATH1210 (calc 1)/Integrals.md +++ b/education/math/MATH1210 (calc 1)/Integrals.md @@ -143,7 +143,7 @@ $$ \text{length of a curve} = \lim_{n \to \infty} \sum_{i=1}^{n}(\sqrt{(x_i - x_ This can also be described as: $$ \text{length of a curve} = \lim_{n \to \infty} \sum_{i=1}^{n}(\sqrt{(\Delta x)^2 +(\Delta y)^2}) $$ 2. Using the mean value theorem: -$$ \lim_{n \to \infty} \sum_{i = 1}^n)\sqrt{\Delta x^2 + (f(x_i) - f(x_{i-1}))i^2} $$ +$$ \lim_{n \to \infty} \sum_{i = 1}^n\sqrt{\Delta x^2 + (f(x_i) - f(x_{i-1}))i^2} $$l $$ \lim_{n \to \infty} \sum_{i=1}^n \sqrt{\Delta x ^2 + (f'(x_{\hat{i}}))(x_i - x_{i-1})^2}$$ 3. Factoring out $\Delta x$ $$ \lim_{n \to \infty} \sum_{i=1}^n \sqrt{\Delta x^2(1 + f'(x_{\hat{i}}))}$$ @@ -196,3 +196,10 @@ $$ L =\int_a^b \sqrt{1 + f'(x)^2} dx$$ 2. Plug into calculator # Area Between Curves +If the area under the curve is found by approximating the space between the curve and the $x$ intercept, then the area *between* two curves can be found by approximating the space between the top curve and the bottom curve. + +Visualized as a set of rectangles, each rectangle would have a corner on the top curve, and a corner on the bottom curve, with a width of $\Delta x$. + +The Riemann Sum definition of the area between two curves is as follows: + +$$ \lim_{n \to \infty} \sum_{i = 0}^n $$ \ No newline at end of file