From c0418af94c35e07d065480585ff2c3f1b7216301 Mon Sep 17 00:00:00 2001 From: arc Date: Tue, 15 Apr 2025 09:21:58 -0600 Subject: [PATCH] vault backup: 2025-04-15 09:21:58 --- education/math/MATH1210 (calc 1)/Integrals.md | 8 +++++++- 1 file changed, 7 insertions(+), 1 deletion(-) diff --git a/education/math/MATH1210 (calc 1)/Integrals.md b/education/math/MATH1210 (calc 1)/Integrals.md index 0c5328a..ffb807f 100644 --- a/education/math/MATH1210 (calc 1)/Integrals.md +++ b/education/math/MATH1210 (calc 1)/Integrals.md @@ -143,4 +143,10 @@ $$ \text{length of a curve} = \lim_{n \to \infty} \sum_{i=1}^{n}(\sqrt{(x_i - x_ This can also be described as: $$ \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) $$ \ No newline at end of file +$$ \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_{\hat{i}}))(x_i - x_{i-1})^2}$$ +3. Factoring out $\Delta x$ +$$ \lim_{n \to \infty} \sum_{i=1}^n \sqrt{x^2(1 + f'(\Delta x_{\hat{i}}))}$$ +4. Moving $\Delta x$ out of the root + +$$ \lim_{n \to \infty} \sum_{i=1}^n \sqrt{x^2(1 + f'(\Delta x_{\hat{i}}))}$$ \ No newline at end of file